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Documenta Math.
Thom Spectra that Are Symmetric Spectra
Christian Schlichtkrull
Received: November 7, 2007
Revised: December 18, 2009
Communicated by Stefan Schwede
Abstract. We analyze the functorial and multiplicative properties
of the Thom spectrum functor in the setting of symmetric spectra and
we establish the relevant homotopy invariance.
2000 Mathematics Subject Classification: 55P43
Keywords and Phrases: Thom spectra, symmetric spectra.
1. Introduction
The purpose of this paper is to develop the theory of Thom spectra in the
setting of symmetric spectra. In particular, we establish the relevant homotopy
invariance and we investigate the multiplicative properties. Classically, given
a sequence of spaces X0 → X1 → X2 → . . . , equipped with a compatible
sequence of maps fn : Xn → BO(n), the Thom spectrum T (f ) is defined by
pulling the universal bundles V (n) over BO(n) back via the fn ’s and letting
T (f )n = f ∗ V (n)/Xn .
Here the bar denotes fibre-wise one-point compactification. More generally,
one may consider compatible families of maps Xn → BF (n), where F (n) is
the topological monoid of base point preserving self-homotopy equivalences
of S n , and similarly define a Thom spectrum by pulling back the canonical S n -(quasi)fibration over BF (n). Composing with the canonical maps
BO(n) → BF (n), one sees that the latter construction generalizes the former. This generalization was suggested by Mahowald [24], [25], and has been
investigated in detail by Lewis in [20].
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1.1. Symmetric Thom spectra via I-spaces. In order to translate the
definition of Thom spectra into the setting of symmetric spectra, we shall
modify the construction by considering certain diagrams of spaces. Let I be
the category whose objects are the finite sets n = {1, . . . , n}, together with the
empty set 0, and whose morphisms are the injective maps. The concatenation
m ⊔ n in I is defined by letting m correspond to the first m and n to the last
n elements of {1, . . . , m + n}. This makes I a symmetric monoidal category
with symmetric structure given by the (m, n)-shuffles τm,n : m ⊔ n → n ⊔ m.
We define an I-space to be a functor from I to the category U of spaces and
write IU for the category of such functors. The correspondence n → BF (n)
defines an I-space and we show that if X → BF is a map of I-spaces, then
the Thom spectrum T (f ) defined as above is a symmetric spectrum. The main
advantage of the category SpΣ of symmetric spectra to ordinary spectra is that
it has a symmetric monoidal smash product. Similarly, the category IU/BF
of I-spaces over BF inherits a symmetric monoidal structure from I and the
Thom spectrum functor is compatible with these structures in the following
sense.
Theorem 1.1. The symmetric Thom spectrum functor T : IU/BF → SpΣ is
strong symmetric monoidal.
That T is strong symmetric monoidal means of course that there is a natural
isomorphism of symmetric spectra T (f ) ∧ T (g) ∼
= T (f ⊠ g), where f ⊠ g denotes
the monoidal product in IU/BF . In particular, T takes monoids in IU/BF to
symmetric ring spectra. A similar construction can be carried out in the setting
of orthogonal spectra and the idea of realizing Thom spectra as “structured ring
spectra” by such a diagrammatic approach goes back to [31].
1.2. Lifting space level data to I-spaces. Let N be the ordered set of
non-negative integers, thought of as a subcategory of I via the canonical subset
inclusions. Another starting point for the construction of Thom spectra is to
consider maps X → BFN , where BFN denotes the colimit of the I-space BF
restricted to N . Given such a map, one may choose a suitable filtration of
X so as to get a map of N -spaces X(n) → BF (n) and the definition of the
Thom spectrum T (f ) then proceeds as above. This is the point of view taken
by Lewis [20]. The space BFN has an action of the linear isometries operad L,
and Lewis proves that if f is a map of C-spaces where C is an operad that is
augmented over L, then the Thom spectrum T (f ) inherits an action of C.
In the setting of symmetric spectra the problem is how to lift space level data
to objects in IU/BF . We think of I as some kind of algebraic structure acting
on BF , and in order to pull such an action back via a space level map we should
ideally map into the quotient space BFI , that is, into the colimit over I. The
problem with this approach is that the homotopy type of BFI differs from
that of BFN . For this reason we shall instead work with the homotopy colimit
BFhI which does have the correct homotopy type. We prove in Section 4 that
the homotopy colimit functor has a right adjoint U : U/BFhI → IU/BF such
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that this pair of adjoint functors defines a Quillen equivalence
hocolim : IU/BF o
I
/ U/BF : U.
hI
Here the model structure on IU is the one established by Sagave-Schlichtkrull
[33]. The weak equivalences in this model structure are called I-equivalences
and are the maps that induce weak homotopy equivalences on the associated
homotopy colimits; see Section 4.1 for details. It follows from the theorem that
the homotopy theory associated to IU/BF is equivalent to that of U/BFhI . As
is often the case for functors that are right adjoints, U is only homotopically
well-behaved when applied to fibrant objects. We shall usually remedy this
by composing with a suitable fibrant replacement functor on U/BFhI and we
write U ′ for the composite functor so defined.
Composing the right adjoint U with the symmetric Thom spectrum functor
from Theorem 1.1 we get a Thom spectrum functor on U/BFhI . However,
even when restricted to fibrant objects this functor does not have all the properties one may expect from a Thom spectrum functor. Notably, one of the
important properties of the Lewis-May Thom spectrum functor on U/BFN is
that it preserves colimits whereas the symmetric Thom spectrum obtained by
composing with U does not have this property. For this reason we shall introduce another procedure for lifting space level data to I-spaces in the form of a
functor
R : U/BFhI → IU/BF
and we shall use this functor to associate Thom spectra to objects in U/BFhI .
The first statement in the following theorem ensures that the functor so defined
produces Thom spectra with the correct homotopy type.
∼
Theorem 1.2. There is a natural level equivalence R −
→ U ′ over BF and the
symmetric Thom spectrum functor defined by the composition
R
T
→ IU/BF −
→ SpΣ
T : U/BFhI −
preserves colimits.
As indicated in the theorem we shall use the notation T both for the symmetric
Thom spectrum on IU/BF and for its composition with R; the context will
always make the meaning clear. In Section 4.4 we show that in a precise sense
our Thom spectrum functor becomes equivalent to that of Lewis-May when
composing with the forgetful functor from symmetric spectra to spectra. We
also have the following analogue of Lewis’ result imposing L-actions on Thom
spectra. In our setting the relevant operad is the Barrat-Eccles operad E, see
[2] and [28], Remarks 6.5. We recall that E is an E∞ operad and that a space
with an E-action is automatically an associative monoid.
Theorem 1.3. The operad E acts on BFhI and if f : X → BFhI is a map of
C-spaces where C is an operad that is augmented over E, then T (f ) inherits an
action of C.
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We often find that the enriched functoriality obtained by working with homotopy colimits over I instead of colimits over N is very useful. For example, one
may represent complexification followed by realification as maps of E-spaces
BOhI → BUhI → BOhI ,
such that the composite E∞ map represents multiplication by 2. The procedure
for lifting space level data described above works quite generally for diagram
categories. Implemented in the framework of orthogonal spectra, it gives an
answer to the problem left open in [32], Chapter 23, on how to construct
orthogonal Thom spectra from space level data; we spell out the details of this
in Section 8.5. We also remark that one can define an I-space BGL1 (A) for
any symmetric ring spectrum A, and that an analogous lifting procedure allows
one to associate A-module Thom spectra to maps X → BGL1 (A)hI . We hope
to return to this in a future paper.
1.3. Homotopy invariance. Ideally, one would like the symmetric Thom
spectrum functor to take I-equivalences of I-spaces over BF to stable equivalences of symmetric spectra. However, due to the fact that quasifibrations
are not in general preserved under pullbacks this is not true without further
assumptions on the objects in IU/BF . We say that an object (X, f ) (that is,
a map f : X → BF ) is T -good if T (f ) has the same homotopy type as the
Thom spectrum associated to a fibrant replacement of f ; see Definition 5.1 for
details.
Theorem 1.4. If (X, f ) → (Y, g) is an I-equivalence of T -good I-spaces over
BF , then the induced map T (f ) → T (g) is a stable equivalence of symmetric
spectra.
Here stable equivalence refers to the stable model structure on SpΣ defined
in [16] and [27]. It is a subtle property of this model structure that a stable equivalence needs not induce an isomorphism of stable homotopy groups.
However, if X and Y are convergent (see Section 2), then the associated Thom
spectra are also convergent, and in this case a stable equivalence is indeed a π∗ isomorphism in the usual sense. The T -goodness requirement in the theorem
is not a real restriction since in general any object in U/BF can (and should)
be replaced by one that is T -good. The functor R takes values in the subcategory of convergent T -good objects and takes weak homotopy equivalences to
I-equivalences (in fact to level-wise equivalences). It follows that the Thom
spectrum functor in Theorem 1.2 is a homotopy functor; see Corollary 4.13.
Example 1.5. Theorem 1.4 also has interesting consequences for Thom spectra that are not convergent. As an example, consider the Thom spectrum
M O(1)∧∞ that represents the bordism theory of manifolds whose stable normal bundle splits as a sum of line bundles, see [1], [9]. This is the symmetric
Thom spectrum associated to the map of I-spaces X(n) → BF (n), where
X(n) = BO(1)n . It is proved in [34] that XhI is homotopy equivalent to
Q(RP ∞ ), hence it follows that M O(1)∧∞ is stably equivalent as a symmetric
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ring spectrum to the Thom spectrum associated to the map of infinite loop
spaces Q(RP ∞ ) → BFhI .
In general any I-space X is I-equivalent to the constant I-space XhI and
consequently any symmetric Thom spectrum is stably equivalent to one arising
from a space-level map. However, the added flexibility obtained by working in
IU is often very convenient. Notably, it is proved in [33] that any E∞ monoid
in IU is equivalent to a strictly commutative monoid; something which is wellknown not to be the case in U.
1.4. Applications to the Thom isomorphism. As an application of the
techniques developed in this paper we present a strictly multiplicative version
of the Thom isomorphism. A map f : X → BFhI gives rise to a morphism in
U/BFhI ,
∆ : (X, f ) → (X × X, f ◦ π2 ),
where ∆ is the diagonal inclusion and π2 denotes the projection onto the second
factor of X × X. The Thom spectrum T (f ◦ π2 ) is isomorphic to X+ ∧ T (f ),
and the Thom diagonal
∆ : T (f ) → X+ ∧ T (f )
is the map of Thom spectra induced by ∆. In Section 7.1 we define a canonical
orientation T (f ) → H, where H denotes (a convenient model of) the EilenbergMac Lane spectrum HZ/2. Using this we get a map of symmetric spectra
(1.6)
∆∧H
T (f ) ∧ H −−−→ X+ ∧ T (f ) ∧ H → X+ ∧ H ∧ H → X+ ∧ H,
where the last map is induced by the multiplication in H. The spectrum level
version of the Z/2-Thom isomorphism theorem is the statement that this is a
stable equivalence, see [26]. If f is oriented in the sense that it lifts to a map
f : X → BSFhI , then we define a canonical integral orientation T (f ) → HZ
and the spectrum level version of the integral Thom isomorphism theorem is
the statement that the induced map
(1.7)
T (f ) ∧ HZ → X+ ∧ HZ
is a stable equivalence. In our framework these results lift to “structured ring
spectra” in the sense of the following theorem. Here C again denotes an operad
that is augmented over E.
Theorem 1.8. If f : X → BFhI (respectively f : X → BSFhI ) is a map of
C-spaces, then the spectrum level Thom equivalence (1.6) (respectively (1.7)) is
a C-map.
For example, one may represent the complex cobordism spectrum M U as the
Thom spectrum associated to the E-map BUhI → BSFhI and the Thom equivalence (1.7) is then an equivalence of E∞ symmetric ring spectra. This should
be compared with the H∞ version in [20].
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1.5. Diagram Thom spectra and symmetrization. The definition of the
symmetric Thom spectrum functor shows that the category I is closely related
to the category of symmetric spectra. However, many of the Thom spectra
that occur in the applications do not naturally arise from a map of I-spaces
but rather from a map of D-spaces for some monoidal category D equipped
with a monoidal functor D → I. We formalize this in Section 8 where we
introduce the notion of a D-spectrum associated to such a monoidal functor.
For example, the complex cobordism spectrum M U associated to the unitary
groups U (n) and the Thom spectrum M B associated to the braid groups B(n)
can be realized as diagram ring spectra in this way. It is often convenient to
replace the D-Thom spectrum associated to a map of D-spaces f : X → BF by
a symmetric spectrum, and our preferred way of doing this is to first transform
f to a map of I-spaces and then evaluate the symmetric Thom spectrum functor
on this transformed map. In this way we end up with a symmetric spectrum
to which we can exploit the structural relationship to the category of I-spaces.
We shall discuss various ways of carrying out this “symmetrization” process
and in particular we shall see how to realize the Thom spectra M U and M B
as (in the case of M U commutative) symmetric ring spectra.
1.6. Organization of the paper. We begin by recalling the basic facts
about Thom spaces and Thom spectra in Section 2, and in Section 3 we introduce the symmetric Thom spectrum functor and show that it is strong symmetric monoidal. The I-space lifting functor R is introduced in Section 4, where
we prove Theorem 1.2 in a more precise form; this is the content of Proposition 4.10 and Corollary 4.13. Here we also compare the Lewis-May Thom
spectrum functor to our construction. We prove the homotopy invariance result Theorem 1.4 in Section 5, and in Section 6 we analyze to what extent the
constructions introduced in the previous sections are preserved under operad
actions. In particular, we prove Theorem 1.3 in a more precise form; this is the
content of Corollary 6.9. The Thom isomorphism theorem is proved in Section
7 and in Section 8 we discuss how to symmetrize other types of diagram Thom
spectra and how the analogue of the lifting functor R works in the context of
orthogonal spectra. Finally, we have included some background material on
homotopy colimits in Appendix A.
1.7. Notation and conventions. We shall work in the categories U and
T of unbased and based compactly generated weak Hausdorff spaces. By a
cofibration we understand a map having the homotopy extension property,
see [39]. A based space is well-based if the inclusion of the base point is a
cofibration. In this paper S n always denotes the one-point compactification of
Rn . By a spectrum E we understand a sequence {En : n ≥ 0} of based spaces
together with a sequence of based structure maps S 1 ∧ En → En+1 . A map of
spectra f : E → F is a sequence of based maps fn : En → Fn that commute
with the structure maps and we write Sp for the category of spectra so defined.
A spectrum is connective if πn (E) = 0 for n < 0 and convergent if there is
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an unbounded, non-decreasing sequence of integers {λn : n ≥ 0} such that the
adjoint structure maps En → ΩEn+1 are (λn + n)-connected for all n.
2. Preliminaries on Thom spaces and Thom spectra
In this section we recall the basic facts about Thom spaces and Thom spectra
that we shall need. The main reference for this material is Lewis’ account in
[20], Section IX. Here we emphasize the details relevant for the construction
of symmetric Thom spectra in Section 3. We begin by recalling the two-sided
simplicial bar construction and some of its properties, referring to [30] for more
details. Given a topological monoid G, a right G-space Y , and a left G-space
X, this is the simplicial space B• (Y, G, X) with k-simplices Y × Gk × X and
simplicial operators


for i = 0
(yg1 , g2 . . . , gk , x),
di (y, g1 , . . . , gk , x) = (y, g1 , . . . , gi gi+1 , . . . , gk , x), for 0 < i < k


(y, g1 , . . . , gk x),
for i = k,
and
si (y, g1 , . . . , gk , x) = (y, . . . , gi−1 , 1, gi , . . . , x), for 0 ≤ i ≤ k.
We write B(Y, G, X) for the topological realization. In the case where X and
Y equal the one-point space ∗, this is the usual simplicial construction of the
classifying space BG. The projection of X onto ∗ induces a map
p : B(Y, G, X) → B(Y, G, ∗)
whose fibres are homeomorphic to X. Furthermore, if X has a G-invariant
basepoint, then the inclusion of the base point defines a section
s : B(Y, G, ∗) → B(Y, G, X).
Recall that a topological monoid is grouplike if the set of components with the
induced monoid structure is a group.
Proposition 2.1 ([19],[30]). If G is a well-based grouplike monoid, then the
projection p is a quasifibration, and if X has a G-invariant base point such that
X is (non-equivariantly) well-based, then the section s is a cofibration.
In general we say that a sectioned quasifibration is well-based if the section
is a cofibration. Let F (n) be the topological monoid of base point preserving
homotopy equivalences of S n , where we recall that the latter denotes the onepoint compactification of Rn . It follows from [19], Theorem 2.1, that this is
a well-based monoid and we let V (n) = B(∗, F (n), S n ). Then BF (n) is a
classifying space for sectioned fibrations with fibre equivalent to S n and the
projection pn : V (n) → BF (n) is a well-based quasifibration. Given a map
f : X → BF (n), let pX : f ∗ V (n) → X be the pull-back of V (n) along f ,
and notice that the section s gives rise to a section sX : X → f ∗ V (n). The
associated Thom space is the quotient space
T (f ) = f ∗ V (n)/sX (X).
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This construction is clearly functorial on the category U/BF (n) of spaces over
BF (n). We often use the notation (X, f ) for an object f : X → BF (n) in
this category. In order for the Thom space functor to be homotopically wellbehaved we would like pX to be a quasifibration and sX to be a cofibration,
but unfortunately this is not true in general. This is the main technical difference compared to working with sectioned fibrations. For our purpose it will
not do to replace the quasifibration pn by an equivalent fibration since we then
loose the strict multiplicative properties of the bar construction required for
the definition of strict multiplicative structures on Thom spectra. We say that
f classifies a well-based quasifibration if pX is a quasi-fibration and sX is a cofibration. The following well-known results are included here for completeness.
Lemma 2.2 ([20]). Given a well-based sectioned quasifibration p : V → B and
a Hurewicz fibration f : X → B, the pullback pX : f ∗ V → X is again a wellbased quasifibration.
Proof. Since f is a fibration the pullback diagram defining f ∗ V is homotopy
cartesian, hence f ∗ V → X is a quasifibration. In order to see that the section
sX is a cofibration, notice that it is the pullback of the section of p along the
Hurewicz fibration f ∗ V → V . The result then follows from Theorem 12 of [40]
which states that the pullback of a cofibration along a Hurewicz fibration is
again a cofibration.
Let Top(n) be the topological group of base point preserving homeomorphisms
of S n . The next result is the main reason why the objects in U/BF (n) that
factor through BTop(n) are easier to handle than general objects.
Lemma 2.3. If f : X → BF (n) factors through BTop(n), then f classifies a
well-based Hurewicz fibration, hence a well-based quasifibration.
Proof. Let W (n) = B(∗, Top(n), S n ). The projection W (n) → BTop(n) is
a fibre bundle by [30], Corollary 8.4, and in particular a Hurewicz fibration.
Suppose that f factors through a map g : X → BTop(n). Then f ∗ V (n) is
homeomorphic to g ∗ W (n) and thus pX is a Hurewicz fibration. We must prove
that the section is a cofibration. Let us use the Strøm model structure [41] on
U to get a factorization g = g2 g1 where g1 is a cofibration and g2 is a Hurewicz
fibration. From this we get a factorization of the pullback diagram defining
g ∗ W (n),
/ W (n)
/ g ∗ W (n)
g ∗ W (n)
2
pX
X
pY
g1
/Y
g2
/ BTop(n),
and it follows from Lemma 2.2 that the section sY of pY is a cofibration. Since
pY is a Hurewicz fibration it follows by the same argument that the induced
map g ∗ W (n) → g2∗ W (n) is also a cofibration. It is clear that the composition
X → g ∗ W (n) → g2∗ W (n) is a cofibration and the conclusion thus follows from
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Lemma 5 of [41], which states that if h = i ◦ j is a composition of maps in
which h and i are both cofibrations, then j is a cofibration as well.
This lemma applies in particular if f factors through BO(n). In order to get
around the difficulty that the Thom space functor is not a homotopy functor
on the whole category U/BF (n) we follow Lewis [20], Section IX, and define a
functor
(2.4)
Γ : U/BF (n) → U/BF (n),
(X, f ) 7→ (Γf (X), Γ(f ))
by replacing a map by a (Hurewicz) fibration in the usual way,
Γf (X) = {(x, ω) ∈ X × BF (n)I : f (x) = ω(0)},
Γ(f )(x, ω) = ω(1).
We sometimes write Γ(X) instead of Γf (X) when the map f is clear from the
context. The natural inclusion X → Γf (X), whose second coordinate is the
constant path at f (x), defines a natural equivalence from the identity functor
on U/BF (n) to Γ. It follows from Lemma 2.2 that the composition of the
Thom space functor T with Γ is a homotopy functor. We think of T (Γ(f )) as
representing the correct homotopy type of the Thom space and say that f is
T -good if the natural map T (f ) → T (Γ(f )) is a weak homotopy equivalence.
In particular, f is T -good if it classifies a well-based quasifibration. It follows
from the above discussion that the restriction of T to the subcategory of T -good
objects is a homotopy functor.
Remark 2.5. The fibrant replacement functors used in [20] and [30] are defined
using Moore paths instead of paths defined on the unit interval I. The use
of Moore paths is less convenient for our purposes since we shall use Γ in
combination with more general homotopy pullback constructions.
The following basic lemma is needed in order to establish the connectivity and
convergence properties of the Thom spectrum functor. It may for example be
deduced from the dual Blakers-Massey Theorem in [14].
Lemma 2.6. Let
V1 −−−−→

p1
y
β
V2

p2
y
B1 −−−−→ B2
be a pullback diagram of well-based quasifibrations p1 and p2 . Suppose that p1
and p2 are n-connected with n > 1 and that β is k-connected. Then the quotient
spaces V1 /B1 and V2 /B2 are well-based and (n − 1)-connected, and the induced
map V1 /B1 → V2 /B2 is (k + n)-connected.
We now turn to Thom spectra. Let N be as in Section 1.2, and write N U for
the category of N -spaces, that is, functors X : N → U. Consider the N -space
BF defined by the sequence of cofibrations
i
i
i
2
1
0
...
BF (2) →
BF (1) →
BF (0) →
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Christian Schlichtkrull
obtained by applying B to the monoid homomorphisms F (n) → F (n + 1) that
take an element u to 1S 1∧u, the smash product with the identity on S 1 . Notice,
that there are pullback diagrams
¯ V (n) −−−−→ V (n + 1)
S1∧




(2.7)
y
y
BF (n)
i
−−−n−→ BF (n + 1),
¯ − denotes fibre-wise smash product with S 1 . Indeed, there clearly
where S 1 ∧
is such a pullback diagram of the underlying simplicial spaces, and topological
realization preserves pullback diagrams. We let N U/BF be the category of
N -spaces over BF . Thus, an object is a sequence of maps
fn : X(n) → BF (n)
that are compatible with the structure maps. Again we may specify the domain
by writing the objects in the form (X, f ).
Definition 2.8. The Thom spectrum functor T : N U/BF → Sp is defined
by applying the Thom space construction level-wise, T (f )n = T (fn ), with
structure maps given by
S 1 ∧ T (fn ) ∼
= T (in ◦ fn ) → T (fn+1 ).
A morphism in N U/BF induces a map of Thom spectra in the obvious way.
As for the Thom space functor, the Thom spectrum functor is not homotopically well-behaved on the whole category N U/BF . We define a functor
Γ : N U/BF → N U/BF by applying the functor Γ level-wise, and we say that
an object (X, f ) is T -good if the induced map T (f ) → T (Γ(f )) is a stable
equivalence. We say that f is level-wise T -good if the induced map is a levelwise equivalence. The following proposition is an immediate consequence of
Lemma 2.2 and Lemma 2.6.
Proposition 2.9. If f : X → BF is T -good, then T (f ) is connective.
An N -space X is said to be convergent if there exists an unbounded, nondecreasing sequence of integers {λn : n ≥ 0} such that X(n) → X(n + 1) is
λn -connected for each n.
Proposition 2.10. If f : X → BF is level-wise T -good and X is convergent,
then T (f ) is also convergent.
Proof. We may assume that f is a level-wise fibration, hence classifies a wellbased quasifibration at each level. If X(n) → X(n + 1) is λn -connected, it
follows from Lemma 2.6 that the structure map S 1 ∧ T (fn ) → T (fn+1 ) is
(λn + n)-connected. The convergence of X thus implies that of T (f ).
Given an N -space X, write XhN for its homotopy colimit. This is homotopy
equivalent to the usual telescope construction on X. We say that a morphism
(X, f ) → (Y, g) in N U/BF is an N -equivalence if the induced map of homotopy
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colimits XhN → YhN is a weak homotopy equivalence. The following theorem
can be deduced from [20], Proposition 4.9. We shall indicate a more direct
proof in Section 5.1.
Theorem 2.11. If (X, f ) → (Y, g) is an N -equivalence of T -good N -spaces
over BF , then the induced map T (f ) → T (g) is a stable equivalence.
In particular, it follows that T ◦ Γ takes N -equivalences to stable equivalences.
3. Symmetric Thom spectra
We begin by recalling the definition of a (topological) symmetric spectrum.
The basic references are the papers [16] and [27] that deal respectively with
the simplicial and the topological version of the theory. See also [35].
3.1. Symmetric spectra. By definition a symmetric spectrum X is a spectrum in which each of the spaces X(n) is equipped with a base point preserving
left Σn -action, such that the iterated structure maps
σ n : S m ∧ X(n) → X(m + n)
are Σm ×Σn -equivariant. A map of symmetric spectra f : X → Y is a sequence
of Σn -equivariant based maps X(n) → Y (n) that strictly commute with the
structure maps. We write SpΣ for the category of symmetric spectra. Following
[27] we shall view symmetric spectra as diagram spectra, and for this reason
we introduce some notation which will be convenient for our purposes. Let
the category I be as in Section 1.1. Given a morphism α : m → n in I, let
n − α denote the complement of α(m) in n and let S n−α be the one-point
compactification of Rn−α . Consider then the topological category IS that has
the same objects as I, but whose morphism spaces are defined by
_
S n−α .
IS (m, n) =
α∈I(m,n)
We view IS as a category enriched in the category of based spaces T . Writing
the morphisms in the form (x, α) for x ∈ S n−α , the composition is defined by
IS (m, n) ∧ IS (l, m) → IS (l, n),
(x, α) ∧ (y, β) 7→ (x ∧ α∗ y, αβ),
where x ∧ α∗ y is defined by the canonical homeomorphism
S n−α ∧ S m−β ∼
= S n−αβ , x ∧ y 7→ x ∧ α∗ y,
obtained by reindexing the coordinates of S m−β via α. This choice of notation
has the advantage of making some of our constructions self-explanatory. By a
functor between categories enriched in T we understand a functor such that
the maps of morphism spaces are based and continuous. Thus, if X : IS → T
is a functor in this sense, then we have for each morphism α : m → n in I a
based continuous map
α∗ : S n−α ∧ X(m) → X(n).
One easily checks that the maps S 1 ∧ X(n) → X(n + 1) induced by the morphisms n → 1 ⊔ n give X the structure of a symmetric spectrum and that the
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category of (based continuous) functors IS → T may be identified with SpΣ
in this way. The symmetric monoidal structure of I gives rise to a symmetric
monoidal structure on IS . On morphism spaces this is given by the continuous
maps
⊔ : IS (m1 , n1 ) × IS (m2 , n2 ) → IS (m1 ⊔ m2 , n1 ⊔ n2 ),
that map a pair of morphisms (x, α), (y, β) to (x ∧ y, α ⊔ β). As noted in
[27], this implies that the category of symmetric spectra inherits a symmetric
monoidal smash product. Given symmetric spectra X and Y , this is defined
by the left Kan extension,
(3.1)
X ∧ Y (n) = colim X(n1 ) ∧ Y (n2 ),
n1 ⊔n2 →n
where the colimit is over the topological category (⊔ ↓ n) of objects and morphisms in IS × IS over n. More explicitly, we may rewrite this as
X ∧ Y (n) =
colim
α : n1 ⊔n2 →n
S n−α ∧ X(n1 ) ∧ Y (n2 ),
where the colimit is now over the discrete category (⊔ ↓ n) of objects and
morphisms in I × I over n. Given a morphism in this category of the form
α′
α
→ n) → (n′1 , n′2 , n′1 ⊔ n′2 −→ n),
(β1 , β2 ) : (n1 , n2 , n1 ⊔ n2 −
the morphism α′ specifies a homeomorphism
′
′
′
S n−α ∼
= S n−α ∧ S n1 −β1 ∧ S n2 −β2 ,
and the induced map in the diagram is defined by
′
′
′
S n−α ∧ X(n1 ) ∧ Y (n2 ) → S n−α ∧ S n1 −β1 ∧ X(n1 ) ∧ S n2 −β2 ∧ Y (n2 )
′
→ S n−α ∧ X(n′1 ) ∧ Y (n′2 ).
The unit for the smash product is the sphere spectrum S with S(n) = S n . By
definition, a symmetric ring spectrum is a monoid in this monoidal category.
Spelling this out using the above notation, a symmetric ring spectrum is a
symmetric spectrum X equipped with a based map 1X : S 0 → X(0) and a
collection of based maps
µm,n : X(m) ∧ X(n) → X(m + n),
such that the usual unitality and associativity conditions hold, and such that
the diagrams
′
′
µm,m′ ◦tw
′
′
S n−α ∧ X(m) ∧ S n −α ∧ X(m′ ) −−−−−−→ S n+n −α⊔α


yα∧α′
X(n) ∧ X(n′ )
µn,n′
−−−−→
∧ X(m + m′ )


yα⊔α′
X(n + n′ )
commute for each pair of morphisms α : m → n and α′ : m′ → n′ in I. Here
′
′
tw flips the factors X(m) and S n −α . A ring spectrum is commutative if it
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defines a commutative monoid in SpΣ . Explicitly, this means that there are
commutative diagrams
µm,n
X(m) ∧ X(n) −−−−→ X(m + n)



τm,n
ytw
y
µn,m
X(n) ∧ X(m) −−−−→ X(n + m),
where the right hand vertical map is given by the left action of the (m, n)-shuffle
τm,n .
3.2. Symmetric Thom spectra via I-spaces. As in Section 1.1 we write
IU for the category of I-spaces. This category inherits the structure of a
symmetric monoidal category from that of I in the usual way: given I-spaces
X and Y , their product X ⊠ Y is defined by the Kan extension
(X ⊠ Y )(n) = colim X(n1 ) × Y (n2 ),
n1 ⊔n2 →n
where the colimit is again over the category (⊔ ↓ n). The unit for the monoidal
structure is the constant I-space I(0, −). We use term I-space monoid for a
monoid in this category. This amounts to an I-space X equipped with a unit
1X ∈ X(0) and a natural transformation of I × I-diagrams
µm,n : X(m) × X(n) → X(m + n)
that satisfies the obvious associativity and unitality conditions. An I-space
monoid X is commutative if it defines a commutative monoid in IU, that is,
the diagrams
µm,n
X(m) × X(n) −−−−→ X(m + n)


τm,n

y
ytw
µn,m
X(n) × X(m) −−−−→ X(n + m)
are commutative.
The family of topological monoids F (n) define a functor from I to the category of topological monoids: a morphism α : m → n in I induces a monoid
homomorphism α : F (m) → F (n) by associating to an element f in F (m) the
composite map
n−α
S
∧f
Sn ∼
= Sn.
= S n−α ∧ S m −−−−−→ S n−α ∧ S m ∼
∼ S n is induced by the bijection
As usual the homeomorphism S n−α ∧ S m =
(n − α) ⊔ m → n specified by α and the inclusion of n − α in n. We also have
the natural monoid homomorphisms
(3.2)
F (m) × F (n) → F (m + n),
(f, g) 7→ f ∧ g
defined by the usual smash product of based spaces. Applying the classifying space functor degree-wise and using that it commutes with products, we
get from this the commutative I-space monoid BF : n 7→ BF (n). We write
IU/BF for the category of I-spaces over BF with objects (X, f ) given by a
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map f : X → BF of I-spaces. This category inherits a symmetric monoidal
structure from that of IU: given objects (X, f ) and (Y, g), the product is
defined by the composition
f ⊠g
f ⊠ g : X ⊠ Y −→ BF ⊠ BF → BF,
where the last map is the multiplication in BF . The meaning of the symbol
f ⊠ g will always be clear from the context. By definition, a monoid in this
monoidal structure is a pair (X, f ) given by an I-space monoid X together
with a monoid morphism f : X → BF .
Definition 3.3. The symmetric Thom spectrum functor T : IU/BF → SpΣ
is defined by the level-wise Thom space construction T (f )(n) = T (fn ). A
morphism α : m → n in I gives rise to a pullback diagram
¯ V (m) −−−−→ V (n)
S n−α ∧




y
y
α
−−−−→ BF (n)
BF (m)
¯ denotes the fibre-wise smash product. On fibres this restricts to the
in which ∧
homeomorphism S n−α ∧ S m → S n specified by α. Pulling this diagram back
via f and applying the Thom space construction, we get the required structure
maps
S n−α ∧ T (fm ) ∼
= T (α ◦ fm ) → T (fn ).
Notice, that this Thom spectrum functor is related to that in Section 2 by a
commutative diagram of functors
T
IU/BF −−−−→ SpΣ




y
y
(3.4)
T
N U/BF −−−−→ Sp,
where the vertical arrows represent the obvious forgetful functors. Recall the
notion of a strong symmetric monoidal functor from [22], Section XI.2. We
now prove Theorem 1.1 stating that the symmetric Thom spectrum is strong
symmetric monoidal.
Proof of Theorem 1.1. It is clear that we have a canonical isomorphism S →
T (∗). We must show that given objects (X, f ) and (Y, g) in IU/BF there is a
natural isomorphism
T (f ) ∧ T (g) ∼
= T (f ⊠ g).
By definition, X ⊠ Y (n) is the colimit of the (⊔ ↓ n)-diagram
α
→ n) 7→ X(n1 ) × Y (n2 ).
(n1 , n2 , n1 ⊔ n2 −
Given α : n1 ⊔ n2 → n, let α(f, g) be the composite map
fn ×gn
α
2
1
−−−→
BF (n1 ) × BF (n2 ) → BF (n1 + n2 ) −
→ BF (n).
X(n1 ) × Y (n2 ) −−−
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Using these structure maps we view the above (⊔ ↓ n)-diagram as a diagram over BF (n), and since the Thom space functor preserves colimits by
[20], Propositions 1.1 and 1.4, we get the homeomorphism
T (f ⊠ g)(n) ∼
= colim T (α(f, g)).
(⊔↓n)
Furthermore, since pullback commutes with topological realization and fibrewise smash products, we have an isomorphism
¯ gn∗ 2 V (n2 )
¯ fn∗1 V (n1 )∧
α(f, g)∗ V (n) ∼
= S n−α ∧
of sectioned spaces over BF (n), hence a homeomorphism of the associated
Thom spaces
T (α(f, g)) ∼
= S n−α ∧ T (fn ) ∧ T (gn ).
1
2
Combining the above, we get the homeomorphism
T (f ) ∧ T (g)(n) ∼
= colim S n−α ∧ T (fn1 ) ∧ T (gn2 ) ∼
= T (f ⊠ g)(n),
α
specifying the required isomorphism of symmetric spectra.
Corollary 3.5. If X is an I-space monoid and f : X → BF a monoid morphism, then T (f ) is a symmetric ring spectrum which is commutative if X
is.
Recall that the tensor of an unbased space K with a symmetric spectrum X
is defined by the level-wise smash product X ∧ K+ . Similarly, the tensor of K
with an I-space X is defined by the level-wise product X × K. For an object
(X, f ) in IU/BF , the tensor is given by (X × K, f ◦ πX ), where πX denotes
the projection onto X. We refer to [6], Chapter 6, for a general discussion of
tensors in enriched categories.
Proposition 3.6. The symmetric Thom spectrum functor preserves colimits
and tensors with unbased spaces.
Proof. The first statement follows [20], Proposition 1.1 and Corollary 1.4, which
combine to show that the Thom space functor preserves colimits. The second
claim is that T (f ◦ πX ) is isomorphic to T (f ) ∧ K+ which follows directly from
the definition.
4. Lifting space level data to I-spaces
The homotopy colimit construction induces a functor
(4.1)
hocolim : IU/BF → U/BFhI ,
I
(X → BF ) 7→ (XhI → BFhI ),
where, given an I-space X, we write XhI for its homotopy colimit. Our first
task in this section is to verify that this is the left adjoint in a Quillen equivalence.
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4.1. The right adjoint of hocolimI . Recall first that the homotopy colimit
functor IU → U has a right adjoint that to a space Y associates the I-space
n 7→ Map(B(n ↓ I), Y ). Here (n ↓ I) denotes the category of objects in
I under n. We refer to [7], Section XII.2.2, and [17] for the details of this
adjunction. The right adjoint in turn induces a functor
U : U/BFhI → IU/BF,
(X, f ) 7→ (Uf (X), U (f ))
by associating to a map f : X → BFhI the map of I-spaces defined by the
upper row in the pullback diagram
Uf (X)


y
U(f )
−−−−→
BF


y
Map(B(− ↓ I), X) −−−−→ Map(B(− ↓ I), BFhI )
The map on the right is the unit of the adjunction. It is immediate that U
is right adjoint to the homotopy colimit functor in (4.1) and we shall prove in
Proposition 4.5 below that the adjunction
/ U/BF : U
(4.2)
hocolim : IU/BF o
I
hI
is a Quillen equivalence when we give U/BFhI the model structure induced by
the Quillen model structure on U and IU/BF the model structure induced by
the I-model structure on IU established by Sagave-Schlichtkrull [33]. Before
describing the I-model structure we recall that IU has a level model structure
in which the weak equivalences and fibrations are defined level-wise. Given
d ≥ 0, let Fd : U → IU be the functor that to a space K associates the I-space
Fd (K) = I(d, −) × K. Thus, Fd is left adjoint to the evaluation functor that
takes an I-space X to X(d). The level structure on IU is cofibrantly generated
with set of generating cofibrations
F I = {Fd (S n−1 ) → Fd (Dn ) : d ≥ 0, n ≥ 0}
obtained by applying the functors Fd to the set I of generating cofibrations
for the Quillen model structure on U. By a relative cell complex in IU we
understand a map X → Y that can be written as the transfinite composition
of a sequence of maps
X = Y0 → Y1 → Y2 → · · · → colim Yn = Y
n≥0
where each map Yn → Yn+1 is the pushout of a coproduct of generating cofibrations. It follows from the general theory for cofibrantly generated model
categories that a cofibration in IU is a retract of a cell complex. We refer the
reader to [15], Section 11.6, for a general discussion of level model structures
on diagram categories.
As explained in Section 1.2, the weak equivalences in the I-model structure
on IU, that is, the I-equivalences, are the maps that induce weak homotopy
equivalences of homotopy colimits. The cofibrations in the I-model structure
are the same as for the level structure and the fibrations can be characterized
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as the maps having the right lifting property with respect to acyclic cofibrations. Again, the I-model structure is cofibrantly generated with F I the set of
generating cofibrations. There also is an explicit description of the generating
acyclic cofibrations and the fibrations but we shall not need this here.
Lemma 4.3. The adjunction in (4.2) is a Quillen adjunction
Proof. We claim that hocolimI preserves cofibrations and acyclic cofibrations.
By definition, hocolimI preserves weak equivalences in general and the first
claim therefore implies the second. For the first claim it suffices to show that
hocolimI takes the generating cofibrations for IU to cofibrations in U. The
homotopy colimit of a map Fd (S n−1 ) → Fd (Dn ) may be identified with the
map
B(d ↓ I) × S n−1 → B(d ↓ I) × Dn
and the claim follows since B(d ↓ I) is a cell complex.
In preparation for the proof that the above Quillen adjunction is in fact a
Quillen equivalence we make some general comments on homotopy colimits of
I-spaces. In general, given an I-space X, the homotopy type of XhI may
be very different from that of XhN . However, if the underlying N -space is
convergent, then the natural map XhN → XhI is a weak homotopy equivalence
by the following lemma due to Bökstedt; see [23], Lemma 2.3.7, for a published
version.
Lemma 4.4 ([8]). Let X be an I-space and suppose that there exists an
unbounded non-decreasing sequence of integers λm such that any morphism
m → n in I induces a λm -connected map X(m) → X(n). Then the inclusion
{n} → I induces a (λn − 1)-connected map X(n) → XhI for all n.
The structure maps F (m) → F (n) are (m − 1)-connected by the Freudentahl
suspension theorem and consequently the induced maps BF (m) → BF (n) are
m-connected. Thus, the proposition applies to the I-space BF and we see that
the canonical map BF (n) → BFhI is (n − 1)-connected. This map can be
written as the composition
BF (n) → Map(B(n ↓ I), BFhI ) → BFhI
where the first map is the unit of the adjunction and the second map is defined
by evaluating at the vertex represented by the initial object. Since the second
map is clearly a homotopy equivalence it follows that the first map is also
(n − 1)-connected.
Proposition 4.5. The adjunction (4.2) is a Quillen equivalence.
Proof. Given a cofibrant object f : X → BF in IU/BF and a fibrant object
g : Y → BFhI in U/BFhI we must show that a morphism φ : XhI → Y of
spaces over BFhI is a weak homotopy equivalence if and only if the adjoint
ψ : X → Ug (Y ) is an I-equivalence of I-spaces over BF . The maps φ and ψ
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are related by the commutative diagram
XhIB
BB
BB
B
φ BB
!
ψhI
Y
/ Ug (Y )hI
v
vv
vvε
v
v
v{ v
where ε denotes the counit for the adjunction. It therefore suffices to show that
ε is a weak homotopy equivalence. The assumption that (Y, g) be a fibrant
object means that g is a fibration and the pullback diagram
Ug (Y )(n)


y
−−−−→
BF (n)


y
Map(B(n ↓ I), Y ) −−−−→ Map(B(n ↓ I), BFhI )
used to define Ug (Y ) is therefore homotopy cartesian. By the remarks following
Lemma 4.4 it follows that the vertical maps are (n − 1)-connected. The counit
ε admits a factorization
Ug (Y )hI → Map(B(− ↓ I), Y )hI → Y
where the first map is a weak homotopy equivalence by the above discussion and
the second map is a weak homotopy equivalence since B(− ↓ I) is level-wise
contractible. This completes the proof.
The functor U is only homotopically well-behaved when applied to fibrant
objects. We define a (Hurewicz) fibrant replacement functor Γ on U/BFhI as
in (2.4) (replacing BF (n) by BFhI ) and we write U ′ for the composite functor
U ◦ Γ. This is up to natural homeomorphism the same as the functor obtained
by evaluating the homotopy pullback instead of the pullback in the diagram
defining Uf (X).
4.2. The I-space lifting functor R. As discussed in Section 1.2, the functor U does not have all the properties one may wish when constructing Thom
spectra from maps to BFhI . In this section we introduce the I-space lifting
functor R and we establish some of its properties. Given a space X and a map
f : X → BFhI , we shall view this as a map of constant I-spaces. In order to
lift it to a map with target BF , consider the I-space BF defined by
BF (n) = hocolim BF ◦ πn ,
(I↓n)
where πn : (I ↓ n) → I is the forgetful functor that maps an object m → n
to m. By definition, BF is the homotopy left Kan extension of BF along
the identity functor on I, see Appendix A.1. Since the identity on n is a
terminal object in (I ↓ n) there results a canonical homotopy equivalence
tn : BF (n) → BF (n) for each n.
Lemma 4.6. The map πn : BF (n) → BFhI induced by the functor πn is (n−1)connected.
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Proof. The homotopy equivalence tn has a section induced by the inclusion of
the terminal object in (I ↓ n), such that the canonical map BF (n) → BFhI
factors through BF (n). The result therefore follows from Lemma 4.4 and the
above discussion.
Consider now the diagram of I-spaces
π
t
− BF −
BFhI ←
→ BF,
where the right hand map is the level-wise equivalence specified above and the
left hand map is induced by the functors πn . Here we again view BFhI as a
constant I-space. We define the I-space Rf (X) to be the level-wise homotopy
pullback of the diagram of I-spaces
f
π
− BF ,
X−
→ BFhI ←
(4.7)
that is, Rf (X)(n) is the space
I
{(x, ω, b) ∈ X × BFhI
× BF (n) : ω(0) = f (x), ω(1) = π(b)}.
Notice, that the two projections Rf (X) → X and Rf (X) → BF are level-wise
Hurewicz fibrations of I-spaces. The functor R is defined by
(4.8)
t
→ BF ).
R : U/BFhI → IU/BF, (f : X → BFhI ) 7→ (R(f ) : Rf (X) → BF −
When there is no risk of confusion we write R(X) instead of Rf (X).
Proposition 4.9. The I-space Rf (X) is convergent and R(f ) is level-wise
T -good.
Proof. Since Rf (X) is defined as a homotopy pullback, we see from Lemma
4.6 that the map Rf (X)(n) → X is (n − 1)-connected for each n, hence Rf (X)
is convergent. We claim that R(f ) classifies a well-based quasifibration at
each level. In order to see this we first observe that t∗ V (n) is a well-based
quasifibration over BF (n) by Lemma A.4. Thus, R(f )∗ V (n) is a pullback of
a well-based quasifibration along the Hurewicz fibration Rf (X)(n) → BF (n),
hence is itself a well-based quasifibration by Lemma 2.2.
∼
Proposition 4.10. There is a natural level-wise equivalence Rf (X) −
→ Uf′ (X)
over BF .
Proof. Given a map f : X → BFhI , consider the diagram of I-spaces
X
f
/ BFhI o
π
BF
t
Map(B(− ↓ I), X)
f
/ Map(B(− ↓ I), BFhI ) o
BF
where we view X and BFhI as constant I-spaces and the corresponding vertical
maps are induced by the projection B(− ↓ I) → ∗. The left hand square is
strictly commutative and we claim that the right hand square is homotopy
commutative. Indeed, with notation as in Appendix A.1, BF is the homotopy
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Kan extension id h∗ BF along the identity functor on I and the adjoints of the
two compositions in the diagram are the two maps hocolimI BF → BFhI
shown to be homotopic in Lemma A.3. Using the canonical homotopy from that
lemma we therefore get a canonical homotopy relating the two composites in the
right hand square. The latter homotopy in turn gives rise to a natural maps of
the associated homotopy pullbacks, that is, to a natural map Rf (X) → Uf′ (X).
Since the vertical maps in the above diagram are level-wise equivalences the
same holds for the map of homotopy pullbacks.
Corollary 4.11. The functors R and hocolimI are homotopy inverses in the
sense that there is a chain of natural weak homotopy equivalences Rf (X)hI ≃ X
of spaces over BFhI and a chain of natural I-equivalences RfhI (XhI ) ≃ X of
I-spaces over BF .
Proof. It follows easily from Proposition 4.5 and its proof that the functor U ′
has this property and the same therefore holds for R by Proposition 4.11. The functor R has good properties both formally and homotopically.
Proposition 4.12. The functor R in (4.8) takes weak homotopy equivalences
over BFhI to level-wise equivalences over BF and preserves colimits and tensors with unbased spaces.
Proof. The first statement follows from the homotopy invariance of homotopy
pullbacks. In order to verify that R preserves colimits, we first observe that
BFhI is locally equiconnected (the diagonal BFhI → BFhI × BFhI is a cofibration) by [19], Corollary 2.4. We then view Rf (X) as the pullback of X
along the level-wise Hurewicz fibrant replacement Γπ (BF ) → BFhI and the
result follows from [20], Propositions 1.1 and 1.2, which together state that
the pullback functor along a Hurewicz fibration preserves colimits provided the
base space is locally equiconnected. The last statement about preservation of
tensors is the claim that if K is an unbased space and (X, f ) an object of
U/BFhI , then R takes (X × K, f ◦ πX ) to Rf (X) × K; this follows immediately
from the definition.
Combining this result with Proposition 2.10, Proposition 3.6 and Proposition
4.9, we get the following corollary in which we define the Thom spectrum
functor on U/BFhI using R.
Corollary 4.13. The Thom spectrum functor
(4.14)
R
T
→ IU/BF −
→ SpΣ
T : U/BFhI −
takes values in the subcategory of well-based, connective and convergent symmetric spectra. It takes weak homotopy equivalences over BFhI to level-wise
equivalences and preserves colimits and tensors with unbased spaces.
The functor R also behaves well with respect to cofibrations as we explain
next. We follow [27] in using the term h-cofibration for a morphism having the
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homotopy extension property. Thus, a map i : A → X in U is an h-cofibration
if and only if the induced map from the mapping cylinder
X ∪i (A × I) → X × I
(4.15)
admits a retraction. By our conventions, this is precisely what we mean by
a cofibration of spaces in this paper. Given a base space B, a morphism i in
U/B is an h-cofibration if the analogous morphism (4.15) admits a retraction
in U/B; we emphasize this by saying that i is a fibre-wise h-cofibration. These
conventions also apply to define h-cofibrations in IU and, given an I-space B,
fibre-wise h-cofibrations in IU/B with the corresponding mapping cylinders
defined level-wise. A morphism i : A → X in SpΣ is an h-cofibration if the
mapping cylinder X ∪i (A ∧ I+ ) is a retract of X ∧ I+ .
Proposition 4.16. The functor R takes maps over BFhI that are cofibrations
in U to fibre-wise h-cofibrations in IU/BF and the Thom spectrum functor
(4.14) takes such maps to h-cofibrations of symmetric spectra.
Proof. Notice first that we may view Rf (X) as the pullback of BF along the
fibrant replacement Γf (X) → BFhI . Given a morphism (A, f ) → (X, g) in
U/BFhI such that A → X is a cofibration, the induced map Γf (A) → Γg (X)
is a fibre-wise h-cofibration by [20], IX, Proposition 1.11. Since fibre-wise hcofibrations are preserved under pullback, this in turn implies that Rf (A) →
Rg (X) is a fibre-wise h-cofibration over BF , hence over BF . It follows from
Proposition 3.6 that the Thom spectrum functor on IU/BF takes fibre-wise hcofibrations to h-cofibrations. Combining this with the above gives the result.
4.3. Preservation of monoidal structures. Recall from [22], Section
XI.2, that given monoidal categories (A, , 1A ) and (B, △, 1B ), a monoidal
functor Φ : A → B is a functor Φ together with a morphism 1B → Φ(1A ) and
a natural transformation
Φ(X)△Φ(Y ) → Φ(XY ),
satisfying the usual associativity and unitality conditions. It follows from the
definition that if A is a monoid in A, then Φ(A) inherits the structure of a
monoid in B. Since (unbased) homotopy colimits commute with products, we
may view hocolimI as a monoidal functor IU → U with structure maps
hocolim X × hocolim Y ∼
= hocolim X × Y → hocolim X ⊠ Y
I
I
I×I
I
induced by the universal natural transformation X(m)×Y (n) → X ⊠Y (m⊔n)
of I × I-diagrams. The unit morphism is induced by the inclusion of the initial
object 0, thought of as a vertex in BI. Since BF is a monoid in IU, BFhI
inherits the structure of a topological monoid. It follows that we may also view
U/BFhI as a monoidal category and the following result is then clear from the
definition.
Proposition 4.17. The functor hocolimI in (4.1) is monoidal.
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However, the functor hocolimI is not symmetric monoidal, hence does not
take commutative monoids in IU to commutative topological monoids. In
particular, BFhI is not a commutative monoid which is already clear from the
fact that it is not equivalent to a product of Eilenberg-Mac Lane spaces. We
prove in Section 6.1 that BFhI has a canonical E∞ structure and that more
generally hocolimI takes E∞ objects in IU to E∞ spaces.
Proposition 4.18. The functor R in (4.8) is monoidal.
Proof. By definition, R(∗)(0) is the loop space of BFhI and we let ∗ → R(∗)
be the map of I-spaces that is the inclusion of the constant loop in degree 0.
We must define an associative and unital natural transformation of I-spaces
R(X) ⊠ R(Y ) → R(X × Y ) over BF . By the universal property of the ⊠product, this amounts to an associative and unital natural transformation of
I 2 -diagrams
R(X)(m) × R(Y )(n) → R(X × Y )(m ⊔ n).
The domain is the homotopy pullback of the diagram
X × Y → BFhI × BFhI ← BF (m) × BF (n),
and the target is the homotopy pullback of the diagram
X × Y → BFhI ← BF (m + n).
The I-space BF inherits a monoid structure from that of BF such that
π : BF → BFhI is a map of monoids. Using these structure maps, we define
a map from the first diagram to the second, giving the required multiplication.
Since the monoids in the monoidal category U/BFhI are precisely the topological monoids over BFhI , this has the following corollary.
Corollary 4.19. If X is a topological monoid and f : X → BFhI a monoid
morphism, then T (f ) is a symmetric ring spectrum.
This may be reformulated as saying that the Thom spectrum functor preserves
the action of the associativity operad whose kth space is the symmetric group
Σk , see [28], Section 3. More generally, we show in Section 6 that T preserves all
operad actions of operads that are augmented over the Barratt-Eccles operad.
4.4. Comparison with the Lewis-May Thom spectrum functor. Let
as before BFN denote the colimit of BF over N . In this section we recall
the Thom spectrum functor on U/BFN considered in [20], Section IX, and we
relate this to our symmetric Thom spectrum functor on U/BFhI . We shall
use the same notation for the I-space BF and its restriction to an N -space.
The colimit functor N U/BF → U/BFN has a right adjoint, again denoted U ,
that to an object f : X → BFN associates the map of N -spaces defined by the
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upper row in the pullback diagram
U(f )
Uf (X) −−−−→ BF




y
y
f
−−−−→ BFN
X
where the vertical map on the right is the unit of the adjunction relating the
colimit and the constant functors. Here we view X and BFN as constant
N -spaces. We again write U ′ for the functor obtained by composing with
the Hurewicz fibrant replacement functor Γ on U/BFN . The Thom spectrum
functor considered in [20] is the composition
U
T
→ N U/BF −
→ Sp,
U/BFN −
where T is the functor from Section 2. (In the language of [20] this is the Thom
prespectrum associated to f . The authors go on to define a spectrum M (f ) with
the property that the adjoint structure maps are homeomorphisms, but this
will not be relevant for the discussion here). The first step in the comparison
to our symmetric Thom spectrum functor on U/BFhI is to relate the spaces
BFN and BFhI . Consider the diagram of weak homotopy equivalences
t
i
→ BFN ,
− BFhN −
BFhI ←
where i is induced from the inclusion i : N → I and t is the canonical projection from the homotopy colimit to the colimit. The former is a weak homotopy equivalence by Lemma 4.4 and the latter is a weak homotopy equivalence since the structure maps are cofibrations. Let us choose a homotopy
inverse j : BFhI → BFhN of i and a homotopy relating i ◦ j to the identity on
BFhI . Here we of course use that these spaces have the homotopy type of a
CW-complex. The precise formulation of the comparison will depend on these
choices. Let ζ be the composite homotopy equivalence
j
t
→ BFN .
→ BFhN −
ζ : BFhI −
In general, given a map φ : B1 → B2 in U, we write φ∗ : U/B1 → U/B2 for the
functor defined by post-composing with φ.
Lemma 4.20. Suppose that φ and ψ are maps from B1 to B2 that are homotopic
by a homotopy h : B1 × I → B2 . Then the functors φ∗ and ψ∗ from U/B1 to
U/B2 are related by a chain of natural weak homotopy equivalences depending
on h.
Proof. Let h∗ : U/B1 → U/B2 be the functor that takes f : X → B1 to
f ×I
h
X × I −−−→ B1 × I −
→ B2 .
The two endpoint inclusions of X in X × I then give rise to the natural weak
homotopy equivalences φ∗ → h∗ ← ψ∗ .
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Applied to the homotopy relating i ◦ j to the identity on BFhI this result gives
a chain of natural weak homotopy equivalences relating the composite functor
j∗
i
∗
U/BFhI
U/BFhI −→ U/BFhN −→
to the identity on U/BFhI .
Lemma 4.21. The two compositions in the diagram
U′
U/BFhI −−−−→ IU/BF


ζ
∗
y∗
yi
U′
U/BFN −−−−→ N U/BF
are related by a chain of natural level-wise equivalences.
Proof. We shall interpolate between these functors by relating both to the N space analogue of the functor U ′ on U/BFhI . Thus, given f : X → BFhN , the
diagram of N -spaces
f
Map(B(− ↓ N ), X) −
→ Map(B(− ↓ N ), BFhN ) ←
− BF
is related by evident chains of term-wise level equivalences to the diagrams
i◦f
Map(i∗ B(− ↓ I), X) −−→ Map(i∗ B(− ↓ I), BFhI ) ←
− BF
and
t◦f
X −−→ BFN ←
− BF.
Evaluating the homotopy pullbacks of these diagrams we get a chain of natural
level-wise equivalences relating the two compositions in the diagram
U/BFN o
t∗
U/BFhN
U′
N U/BF o
i∗
/ U/BFhI
U′
∗
i
IU/BF.
By the remarks following Lemma 4.20 we therefore get a chain of natural transformations
(4.22)
i∗ ◦ U ′ ∼ i∗ ◦ U ′ ◦ i∗ ◦ j∗ ∼ U ′ ◦ t∗ ◦ j∗ ∼ U ′ ◦ ζ∗ ,
each of which is a level-wise weak homotopy equivalence.
We can now compare our symmetric Thom spectrum functor to the LewisMay Thom spectrum functor on U/BFN . Since the functors T R and T U ′ on
U/BFhI are level-wise equivalent by Proposition 4.10, it suffices to consider
T U ′.
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Proposition 4.23. The two compositions in the diagram
T U′
U/BFhI −−−−→ SpΣ



ζ
y
y∗
T U′
U/BFN −−−−→ Sp
are related by a chain of level-wise equivalences.
Proof. The diagram in question is obtained by composing the diagram in
Proposition 4.21 with the commutative diagram (3.4). Since the chain of weak
homotopy equivalences in (4.22) is contained in the full subcategory of levelwise T -good objects in N U/BF , applying T gives a chain of level-wise equivalences.
5. Homotopy invariance of symmetric Thom spectra
In this section we prove the homotopy invariance result stated in Theorem 1.4
and we show how the proof can be modified to give the N -space analogue in
Theorem 2.11. As for the Thom space functor, the symmetric Thom spectrum functor is not homotopically well-behaved on the whole domain category IU/BF . We define a level-wise Hurewicz fibrant replacement functor on
IU/BF by applying the functor Γ in (2.4) at each level.
Definition 5.1. An object (X, f ) in IU/BF is T -good if the canonical map
T (f ) → T (Γ(f )) is a stable equivalence (a weak equivalence in the stable model
structure) of symmetric spectra.
As before we say that (X, f ) is level-wise T -good if T (f ) → T (Γ(f )) is a levelwise equivalence. The first step in the proof of Theorem 1.4 is to generalize the
definition of BF to any I-space X by associating to X the I-space X defined
by
X(n) = hocolim X ◦ πn .
(I↓n)
We then have a diagram of I-spaces
π
t
→ X,
−X−
XhI ←
where we view XhI as a constant I-space. The map t is a level-wise equivalence
and π is an I-equivalence by Lemma A.2. If f : X → BF is a map of I-spaces,
then we have a commutative diagram
f¯
X −−−−→

π
y
fhI
BF

π
y
XhI −−−−→ BFhI ,
hence there is an induced morphism
(5.2)
(X, t ◦ f¯) → (RfhI (XhI ), R(fhI ))
of I-spaces over BF .
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Proposition 5.3. Applying T ◦ Γ to the morphism (5.2) gives a stable equivalence of symmetric spectra.
In order to prove this proposition we shall make use of the level model structure
on IU recalled in Section 4.1. Let Fd : U → IU be the functor defined in that
section and let us write Fd (u) : Fd (K) → BF for the map of I-spaces associated
to a map of spaces u : K → BF (d).
Lemma 5.4. If u is a Hurewicz fibration, then Fd (u) is level-wise T -good.
Proof. The pullback of V (n) along Fd (u) is isomorphic to the coproduct of the
¯ V (d) over BF (d), where
pullbacks along u of the fibre-wise suspensions S n−α ∧
α runs through the injective maps d → n. These are well-based quasifibrations
by Proposition 2.1 and since u is a fibration, the same holds for the pullbacks
by Lemma 2.2 and the claim follows.
The idea is to first prove Proposition 5.3 for objects of the form Fd (u).
Lemma 5.5. Applying T ◦ Γ to the map of I-spaces Fd (K) → R(Fd (K)hI ) over
BF gives a stable equivalence of symmetric spectra.
The proof of this requires some preparation. We view K as a space over BFhI
via the map
(5.6)
ũ : K → BF (d) → BFhI ,
where the second map is induced by the inclusion of {d} in I.
Lemma 5.7. There is a weak homotopy equivalence K → Fd (K)hI of spaces
over BFhI .
Proof. By definition of the homotopy colimit we may identify Fd (K)hI with
B(d ↓ I) × K, where (d ↓ I) is the category of objects in I under d. Since
this category has an initial object its classifying space is contractible and the
result follows.
In the case of the I-space Fd (K), the level-wise equivalence t : Fd (K) → Fd (K)
has a section induced by the canonical map K → Fd (K)(d). Using this, we get
a commutative diagram in IU/BF ,
∼
(5.8)
Fd (K) −−−−→


y
∼
Fd (K)


y
R(K) −−−−→ R(Fd (K)hI ).
The upper horizontal map is a level-wise equivalence since t is and the lower
horizontal map is a level-wise equivalence by the above lemma. Thus, in order
to prove Lemma 5.5, we may equally well consider the vertical map on the left
hand side of the diagram.
Given a based space T , let FdS (T ) be the symmetric spectrum IS (d, −) ∧ T .
The functor FdS so defined is left adjoint to the functor SpΣ → T that takes a
symmetric spectrum to its dth space, see [27]. In particular it follows from the
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definition that F0S (T ) is the suspension spectrum of T . Notice also that the
Thom spectrum T (Fd (u)) associated to Fd (u) may be identified with FdS (T (u)),
where as usual T (u) denotes the Thom space of the map u. Let T (ũ) be the
symmetric Thom spectrum of the map ũ in (5.6) and let ΣdL T (ũ) be the left
shift by d, that is, the composition of T (ũ) with the concatenation functor
IS → IS , n 7→ d ⊔ n. Thus, the nth space of ΣdL T (ũ) is T (ũ)(d ⊔ n) with Σn
acting via the inclusion Σn → Σd+n induced by n 7→ d ⊔ n. The condition that
u be a Hurewicz fibration in the following lemma is unnecessarily restrictive,
but the present formulation is sufficient for our purposes.
Lemma 5.9. If u is a Hurewicz fibration, then the canonical map of spaces
T (u) → T (ũ)(d) induces a π∗ -isomorphism F0S (T (u)) → ΣdL T (ũ).
Proof. In spectrum degree n this is the map of Thom spaces induced by the
map
K → Rũ (K)(d) → Rũ (K)(d ⊔ n),
viewed as a map of T -good spaces over BF (d + n). This is also a map of spaces
over K via the projection Rũ (K) → K, and it therefore follows from the proof
of Proposition 4.9 that its connectivity tends to infinity with n. The result
then follows from Lemma 2.6.
We shall prove Lemma 5.5 using the detection functor D from [37]. We recall
that this functor associates to a symmetric spectrum T the symmetric spectrum
DT whose nth space is the based homotopy colimit
DT (n) = hocolim Ωm (T (m) ∧ S n ).
m∈I
By [37], Theorem 3.1.2, a map of (level-wise well-based) symmetric spectra
T → T ′ is a stable equivalence if and only if the induced map DT → DT ′ is a
π∗ -isomorphism. There is a closely related functor T 7→ M T , where M T is the
symmetric spectrum with nth space
M T (n) = hocolim Ωm (T (m ⊔ n)).
m∈I
Thus, M T is the homotopy colimit of the I-diagram of symmetric spectra
m 7→ Ωm (Σm
L T ). There is a canonical map DT → M T , which is a level-wise
equivalence if T is convergent and level-wise well-based.
Proof of Lemma 5.5. We claim that applying T ◦ Γ to the vertical map on the
left hand side of (5.8) gives a stable equivalence, and for this we may assume
without loss of generality that u is a Hurewicz fibration. Then Fd (u) is T -good
by Lemma 5.4 and since R(ũ) is T -good by Proposition 4.9, it suffices to show
that T (Fd (u)) → T (ũ) is a stable equivalence. Furthermore, by [27], Theorem
8.12, it is enough to show that this map is a stable equivalence after smashing
with S d and by the above remarks this in turn follows if applying D gives a
π∗ -isomorphism. We identify T (Fd (u)) with FdS (T (u)) and claim that there is
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a commutative diagram
F0S (T (u))

∼
y
∼
−−−−→
ΣdL T (ũ)
∼
−−−−→ Ωd (S d ∧ ΣdL T (ũ))

∼
y
∼
D(S d ∧ FdS (T (u))) −−−−→ D(S d ∧ T (ũ)) −−−−→
M (S d ∧ T (ũ)),
where the maps are π∗ -isomorphisms as indicated. The vertical map on the
left hand side is induced by the space-level map
T (u) → FdS (T (u))(d) → Ωd (S d ∧ FdS (T (u))(d)) → D(S d ∧ FdS (T (u)))(0).
It is a fundamental property of the model structure on SpΣ that the induced
map of symmetric spectra is a π∗ -isomorphism, see the proof of [37], Lemma
3.2.5. The first map in the upper row is the stable equivalence from Lemma
5.9, and the remaining indicated arrows are π∗ -isomorphisms since T (ũ) is
connective and convergent. This proves the claim.
We now wish to prove Proposition 5.3 by an inductive argument based on the
filtration
(5.10)
∅ = X0 → X1 → X2 → · · · → colim Xn = X
n
of a cell complex X in IU, cf. the discussion of the level model structure in
Section 4.1. In order to carry out the induction step, we need to ensure that the
induced maps of Thom spectra are h-cofibrations in the sense of Section 4.2.
The following is the I-space analogue of [20], IX, Lemma 1.9 and Proposition
1.11. The proof is essentially the same as in the space-level case.
Proposition 5.11. The functor Γ on IU/BF preserves colimits and takes
morphisms in IU/BF that are h-cofibrations in IU to fibre-wise h-cofibrations.
Since the symmetric Thom spectrum functor on IU/BF preserves colimits and
takes fibre-wise h-cofibrations to h-cofibrations by Proposition 3.6, this has the
following consequence.
Proposition 5.12. The composite functor T ◦ Γ preserves colimits and takes
morphisms in IU/BF that are h-cofibrations in IU to h-cofibrations of symmetric spectra.
Proof of Proposition 5.3. Using the level model structure we may choose a cofibrant I-space X ′ and a level-wise equivalence X ′ → X, hence it suffices to
consider the case where X is a cofibrant I-space. Then X is a retract of a cell
complex which we may view as a cell complex over BF via the retraction. By
functoriality we are thus reduced to the case where X is a cell complex with
a filtration by h-cofibrations as in (5.10). In order to handle this case we use
that both functors in (5.2) preserve colimits and tensors with unbased spaces,
hence they also preserve (not necessarily fibre-wise) h-cofibrations. Applying
the functor T ◦ Γ, we see that both functors in the proposition preserve colimits and take h-cofibrations of I-spaces over BF to h-cofibrations of symmetric
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spectra. We prove by induction that the result holds for each of the I-spaces
Xn in the filtration. By definition, Xn+1 is a pushout of a diagram of the form
B ← A → Xn , where A → B is a coproduct of generating cofibrations, hence
in particular an h-cofibration. We view this as a diagram of I-spaces over BF
via the inclusion of Xn+1 in X and get a diagram of Thom spectra
T Γ(X n )


y
←−−−−
T Γ(A)


y
−−−−→
T Γ(B)


y
T Γ(R((Xn )hI )) ←−−−− T Γ(R(AhI )) −−−−→ T Γ(R(BhI )),
such that the map for Xn+1 is the induced map of pushouts. By the above
discussion it follows that the horizontal maps on the right hand side of the
diagram are h-cofibrations and the vertical maps are stable equivalences by
Lemma 5.5 and the induction hypothesis. Consequently the map of pushouts
is also a stable equivalence, see [27], Theorem 8.12.
Proof of Theorem 1.4. We prove that applying the functor T ◦ Γ to an Iequivalence X → Y over BF gives a stable equivalence of symmetric spectra.
Consider the commutative diagram
X ←−−−−


y
X −−−−→ R(XhI )




y
y
Y ←−−−− Y −−−−→ R(YhI )
of I-spaces over BF . Applying T ◦ Γ to this diagram we get a diagram of
symmetric spectra where the horizontal maps are stable equivalence by Proposition 5.3 and the fact that T ◦ Γ preserves level-wise equivalences. The result
now follows from Corollary 4.13 which ensures that the map R(XhI ) → R(YhI )
induces a stable equivalence.
Notice, that as a consequence of the theorem, the composite functor T ◦ Γ is a
homotopy functor on IU/BF in the sense that it takes I-equivalences to stable
equivalences.
5.1. The proof of Theorem of 2.11. The proof of Theorem 2.11 is similar
to but simpler than the proof of Theorem 1.4. We first introduce a functor
RN : U/BFhN → N U/BF,
which is the N -space analogue of the functor R. Let us temporarily write BF
for the homotopy Kan extension of the N -space BF along the identity functor
of N , that is,
BF (n) = hocolim BF ◦ πn ,
(N ↓n)
where πn is the forgetful functor (N ↓ n) → N . Given a map f : X → BFhN ,
we define RfN (X) to be the level-wise homotopy pullback of the diagram of
N -spaces
f
π
− BF ,
X−
→ BFhN ←
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and we define RN (f ) to be the composite map of N -spaces
t
RN (f ) : RfN (X) → BF −
→ BF.
Exactly as in the I-space case there is a map of N -spaces
(5.13)
(X, t ◦ f ) → (RfNhN (XhN ), RN (fhN ))
over BF , where we again use the (temporary) notation X for the homotopy
Kan extension along the identity on N . Theorem 2.11 then follows from the
following proposition in the same way that Theorem 1.4 follows from Proposition 5.3.
Proposition 5.14. Applying T ◦ Γ to (5.13) gives a stable equivalence of spectra.
In order to prove this we first consider the N -spaces Fd (K) defined by
N (d, −) × K, where d is an object in N and K is a space. Given a map
u : K → BF (d), we have the following N -space analogue of (5.8),
Fd (K) −−−−→


y
Fd (K)


y
R(K) −−−−→ R(Fd (K)hN ).
However, in contrast to the I-space setting, this is a diagram of convergent
N -spaces and the connectivity of the maps in degree n tends to infinity with
n. Thus, the N -space analogue of Lemma 5.5 holds with a simpler proof.
Proof of Proposition 5.14. We use that N U has a cofibrantly generated level
model structure and as in the I-space case we reduce to the case of a cell
complex. Using that the functors in (5.13) preserve colimits and h-cofibrations,
the inductive argument used in the proof of Proposition 5.3 then also applies
in the N -space setting.
6. Preservation of operad actions
Let C be an operad as defined in [28] and notice that C defines a monad C on
the symmetric monoidal category IU in the usual way by letting
∞
a
C(k) ×Σk X ⊠k .
C(X) =
k=0
We define a C-I-space to be an algebra for this monad and write IU[C] for
the category of such algebras. More explicitly, a C-I-space is an I-space X
together with a sequence of maps of I-spaces
θk : C(k) × X ⊠k → X,
satisfying the associativity, unitality and equivariance relations listed in [28],
Lemma 1.4. By the universal property of the ⊠-product, θk is determined by
a natural transformation of I k -diagrams
(6.1)
θk : C(k) × X(n1 ) × · · · × X(nk ) → X(n1 + · · · + nk )
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and the equivariance condition amounts to the commutativity of the diagram
θk ◦(σ×id)
C(k) × X(n1 ) × · · · × X(nk )


yid×σ
−−−−−−→
C(k) × X(nσ−1 (1) ) × · · · × X(nσ−1 (k) )
θ
−−−k−→
X(n1 + · · · + nk )

σ(n ,...,n )
k ∗
y 1
X(nσ−1 (1) + · · · + nσ−1 (k) )
for all elements σ in Σk . Here σ permutes the factors on the left hand side of
the diagram and σ(n1 , . . . , nk ) denotes the permutation of n1 ⊔ · · · ⊔ nk that
permutes the k summands as σ permutes the elements of k. As defined in [28],
the 0th space of C is a one-point space, so that θ0 specifies a base point of X.
Notice, that an action of the one-point operad ∗ on an I-space X is the same
thing as a commutative monoid structure on X. In this case the projection
C → ∗ induces a C-action on X for any operad C. This applies in particular to
the commutative I-space monoid BF .
In similar fashion an operad C defines a monad C on the category SpΣ by
letting
∞
_
C(X) =
C(k)+ ∧Σk X ∧k
k=0
Σ
and we write Sp [C] for the category of algebras for this monad. Thus, an
object of SpΣ [C] is a symmetric spectrum X together with a sequence of maps
of symmetric spectra
θk : C(k)+ ∧ X ∧k → X,
satisfying the analogous associativity, unitality and equivariance relations. By
the universal property of the smash product, θk is determined by a natural
transformation of ISk -diagrams,
(6.2)
θk : C(k)+ ∧ X(n1 ) ∧ · · · ∧ X(nk ) → X(n1 + · · · + nk ).
The naturality condition can be formulated explicitly as follows. Given a family
of morphisms αi : mi → ni in I for i = 1, . . . , k, let α = α1 ⊔ · · · ⊔ αk . Writing
n = n1 + · · · + nk and making the identification
S n1 −α1 ∧ · · · ∧ S nk −αk = S n−α ,
we require that the diagram
S n−α ∧θ
k
S n−α ∧ C(k)+ ∧ X(m1 ) ∧ · · · ∧ X(mk ) −−−−−−→
S n−α ∧ X(m1 + · · · + mk )




y
y
C(k)+ ∧ X(n1 ) ∧ · · · ∧ X(nk )
θ
−−−k−→
X(n1 + · · · + nk )
be commutative. We now show that the symmetric Thom spectrum functor
behaves well with respect to operad actions. Given an operad C and a map of
I-spaces f : X → BF , let C(f ) be the composite map
C(f ) : C(X) → C(BF ) → BF.
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The following is the analogue in our setting of [20], Theorem IX 7.1. It is a
formal consequence of the fact that T is a strong symmetric monoidal functor
that preserves colimits and tensors with unbased spaces.
Proposition 6.3. There is a canonical isomorphism of symmetric spectra
T (C(f )) = C(T (f )).
Corollary 6.4. The Thom spectrum functor on IU/BF preserves operad
actions in the sense that there is an induced functor
T : IU[C]/BF → SpΣ [C].
6.1. Operad actions preserved by hocolimI . As in Section 1.2 we use the
notation E for the Barratt-Eccles operad. We recall that the kth space E(k)
is the classifying space of the translation category Σ̃k that has the elements of
Σk as its objects. A morphism ρ : σ → τ in Σ̃k is an element ρ ∈ Σk such that
ρσ = τ ; see [29], Section 4 (but notice that the order of the composition in Σ̃k
is defined differently here). In the following proposition, C denotes an arbitrary
operad and E × C denotes the product operad whose kth space is the product
E(k) × C(k).
Proposition 6.5. The functor hocolimI induces a functor
hocolim : IU[C] → U[E × C].
I
Proof. Let I(X) be the topological category whose space of objects is the
disjoint union of the spaces X(n) indexed by the objects n in I, and in which a
morphism (m, x) → (n, y) is specified by a morphism α : m → n in I such that
α∗ (x) = y. Then it follows from the definition of the homotopy colimit that
XhI may be identified with the classifying space BI(X); see Appendix A for
details. In the following we shall view the spaces C(k) as topological categories
with only identity morphisms. For each k, consider the functor of topological
categories
ψk : Σ̃k × C(k) × I(X)k → I(X),
that maps a tuple of objects σ ∈ Σk , c ∈ C(k), and (n1 , x1 ), . . . , (nk , xk ), to
(nσ−1 (1) ⊔ · · · ⊔ nσ−1 (k) , σ(n1 , . . . , nk )∗ θk (c, x1 , . . . , xk )).
Here θk denotes the C(k)-action on X and σ(n1 , . . . , nk ) is defined as at the
beginning of this section. If ρ : σ → τ is a morphism in Σ̃k , then the induced
morphism in I(X) is specified by
ψk (ρ) = ρ(nσ−1 (1) , . . . , nσ−1 (k) ),
and if α
~ denotes a k-tuple of morphisms in I(X) whose ith component is
specified by αi : ni → mi , then the induced morphism in I(X) is specified by
ψk (~
α) = ασ−1 (1) ⊔ · · · ⊔ ασ−1 (k) .
Since the classifying space functor preserves products, these functors give rise
to maps
ψk : E(k) × C(k) × BI(X)k → BI(X),
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and it is straightforward to check that this defines an E × C-action on BI(X).
The associativity, unitality, and equivariance conditions may all be checked on
the categorical level.
Letting C be the commutativity operad ∗, it follows in particular that if X is a
commutative I-space monoid, then XhI inherits an E-action. In this case the
action is induced by a permutative structure on the category I(X) introduced
in the above proof, cf. [29], Section 4. This applies in particular to the I-space
BF giving an E-action on BFhI . We say that an operad C is augmented over E
if there is a specified morphism of operads C → E. In this case we may restrict
an (E × C)-action to the diagonal C-action via the morphism C → E × C.
Corollary 6.6. If C is augmented over E, then hocolimI induces a functor
hocolim : IU[C]/BF → U[C]/BFhI .
I
6.2. Operad actions preserved by R. In order to prove that the I-space
lifting functor R preserves operad actions, we need the following lemma in
which we view BFhI as a constant E-I-space.
Lemma 6.7. The I-space BF has an E-action such that π : BF → BFhI is a
morphism of E-I-spaces.
Proof. Consider more generally a commutative I-space monoid X, and let X
be the I-space defined in Section 5. For each object n in I, let I/n(X) be the
topological category whose classifying space is X(n). Thus, the object space is
given by
a
X(m),
α : m→n
where the coproduct is over the objects in (I ↓ n); see Appendix A for details.
Consider for each k the functor
ψk : Σ̃k × I/n1 (X) × · · · × I/nk (X) → I/(n1 ⊔ · · · ⊔ nk )(X)
that maps a tuple of objects σ in Σ̃k and (αi , xi ) in I/ni (X) for i = 1, . . . , k,
to the object
(α, xσ−1 (1) . . . xσ−1 (k) ),
where α is the morphism
ασ−1 (1) ⊔···⊔ασ−1 (k)
α : mσ−1 (1) ⊔ · · · ⊔ mσ−1 (k) −−−−−−−−−−−−−→ nσ−1 (1) ⊔ · · · ⊔ nσ−1 (k)
σ−1 (nσ−1 (1) ,...,nσ−1 (k) )
−−−−−−−−−−−−−−−−→ n1 ⊔ · · · ⊔ nk
and the second factor is the product of the elements xσ−1 (1) , . . . , xσ−1 (k) using
the monoid structure. The induced maps of classifying spaces
ψk : E(k) × X(n1 ) × · · · × X(nk ) → X(n1 + · · · + nk )
then specify the required E-action on X. With this definition it is clear that
the canonical morphism X → XhI is a morphism of E-I-spaces.
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Proposition 6.8. Let C be an operad augmented over the Barratt-Eccles operad. Then the I-space lifting functor R induces a functor
R : U[C]/BFhI → IU[C]/BF.
Proof. We give BFhI the C-action defined by the augmentation to E. Let
f : X → BFhI be a map of C-spaces and consider the diagram
f
π
− BF
X−
→ BFhI ←
defining Rf (X). Pulling the E-action on BF defined in Lemma 6.7 back to a Caction, this is a diagram of C-I-spaces. Thus, Rf (X) is a homotopy pullback of
a diagram in IU[C], hence is itself an object in this category and the projections
Rf (X) → X and Rf (X) → BF are maps of C-I-spaces, see [28], Section 1.
Since the equivalence t : BF → BF is also a map of C-I-spaces, the conclusion
follows.
Combining this with Corollary 6.4 we get the following.
Corollary 6.9. If C is an operad that is augmented over E, then the Thom
spectrum functor on U/BFhI induces a functor T : U[C]/BFhI → SpΣ [C]. 7. The Thom isomorphism
Let M F be the symmetric Thom spectrum associated to the identity BF →
BF , and let M SF be the symmetric Thom spectrum associated to the inclusion
BSF → BF . Here SF (n) denotes the submonoid of orientation preserving
based homotopy equivalences (those that are homotopic to the identity) and
BSF is the corresponding commutative I-space monoid. We first construct
canonical orientations of these Thom spectra, and for this we need convenient
models of Eilenberg-Mac Lane spectra.
7.1. Eilenberg-Mac Lane spectra and orientations. Let A be a discrete ring, and write A[−] for the functor that to a topological space X associates the free topological A-module A[X] generated by X, see e.g. [42], Section
2.3. In the special case where X is the realization of a simplicial set X• this
may be identified with the realization of the simplicial A-module A[X• ]. If X
is based, we write A(X) for the topological A-module A[X]/A[∗]. The functor
A(−) defined in this way is left adjoint to the forgetful functor from topological A-modules to based spaces. It is well-known that A(S n ) is a model of the
Eilenberg-Mac Lane space K(A, n), and that when equipped with the obvious
structure maps this defines a model of the Eilenberg-Mac Lane spectrum for
A as a symmetric ring spectrum. In order to define the orientations, we shall
consider a variant of this construction. Let FA (n) be the topological monoid
of continuous A-linear endomorphisms of A(S n ) and notice that by the above
remarks this is homotopy equivalent to A considered as a discrete multiplicative monoid. Writing SFA (n) for the connected component corresponding to
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the unit of A, this is then a contractible topological monoid. Applying the bar
construction as in Section 2, we get a well-based quasifibration
B(∗, SFA (n), A(S n )) → BSFA (n),
and we define the Eilenberg-Mac Lane spectrum HA to be the symmetric
spectrum with nth space
HA(n) = B(∗, SFA (n), A(S n ))/BSFA (n).
It is easy to check that this is a commutative symmetric ring spectrum which is
level-wise equivalent to the usual model for the Eilenberg-Mac Lane spectrum
considered above. Since HA is flat in the sense of [4], the functor HA ∧ (−)
preserves stable equivalences between well-based spectra; this follows from a
slight refinement of the argument used in [4]. (Alternatively, one can check
that the arguments in [35], Proposition 5.14, works equally well with Quillen
cofibrations of spaces replaced by our notion of (h-)cofibrations.) Let now
A = Z/2 and observe that the functor Z/2(−) defines a map of sectioned
quasifibrations
B(∗, F (n), S n ) → B(∗, SFZ/2 (n), Z/2(n)).
The canonical orientation of M F is the induced map of commutative symmetric
ring spectra M F → HZ/2. Similarly, the functor Z(−) defines a map of
sectioned quasifibrations
B(∗, SF (n), S n ) → B(∗, SFZ (n), Z(S n ))
and the canonical orientation of M SF is the induced map of commutative
symmetric ring spectra M SF → HZ.
7.2. The Thom isomorphism. We first consider the Thom isomorphism with
Z/2-coefficients. Given a map f : X → BFhI , the I-space lift Rf (X) → BF
induces a map of symmetric spectra T (f ) → M F , and we define the HZ/2orientation of T (f ) to be the composition
T (f ) → M F → HZ/2.
As explained in Section 1.4, the orientation induces a map of symmetric spectra
(7.1)
T (f ) ∧ HZ/2 → X+ ∧ HZ/2.
Since our construction of the Thom spectrum functor has good properties both
formally and homotopically, the proof that this is a stable equivalence is almost
completely formal.
Theorem 7.2. The map of symmetric spectra (7.1) is a stable equivalence.
Proof. Both functor in the theorem are homotopy functors on U/BFhI in the
sense that they take weak homotopy equivalences to stable equivalences; this
follows from Corollary 4.13 and the fact that HZ/2 is flat. Thus, we may
assume that X is a CW-complex and consider the filtration of X by skeleta
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X n such that X −1 is the empty set and X n is homeomorphic to the pushout
of a diagram of the form
a
a
X n−1 ←
S n−1 →
Dn .
Since both functors in the theorem preserve pushouts and h-cofibrations, it
suffices by [27], Theorem 8.12, to consider the case where the domain of f is of
the form Dn or S n . If f is the inclusion of the basepoint ∗ → BFhI , then the
unit of the I-space monoid Rf (∗) gives a stable equivalence S → T (f ) and the
composition
∼
S ∧ HZ/2 −
→ T (f ) ∧ HZ/2 → ∗+ ∧ HZ/2
is the identity on HZ/2. Using the homotopy invariance of the Thom spectrum
functor, this easily implies the result for Dn . Identifying S n with the pushout of
the diagram Dn ← S n−1 → Dn , the result for S n then follows by an inductive
argument.
7.3. The integral Thom isomorphism. Using the commutative I-space
monoid BSF instead of BF , we get a a monoidal I-space lifting functor
R : U/BSFhI → IU/BSF
defined in analogy with the I-space lifting functor on U/BFhI . The two lifting
functors are related by a diagram
R
U/BSFhI −−−−→ IU/BSF




y
y
R
U/BFhI −−−−→ IU/BF,
which is commutative up to natural I-equivalence. Thus, the two natural ways
to define a Thom spectrum functor on U/BSFhI are equivalent up to stable
equivalence. For the definition of orientations it is most convenient to define
the Thom spectrum functor on U/BSFhI to be the composition
R
T
→ IU/BSF → IU/BF −
→ SpΣ .
T : U/BSFhI −
With this definition we have a canonical integral orientation of the Thom
spectrum associated to a map X → BSFhI , defined by the composition
T (f ) → M SF → HZ. The orientation again gives rise to a map of symmetric
spectra
(7.3)
T (f ) ∧ HZ → X+ ∧ HZ
and the proof of the integral version of the Thom isomorphism theorem is
completely analogous to the HZ/2-version.
Theorem 7.4. The map (7.3) is a stable equivalence.
We can now verify the claim in Theorem 1.8 that the Thom equivalence is
strictly multiplicative.
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Proof of Theorem 1.8. Let H denote either one of the commutative symmetric
ring spectra HZ/2 or HZ, and view H as an object in SpΣ [C] by projecting
C onto the commutativity operad. We claim that (1.6) is a diagram in SpΣ [C]
when we give each of the terms the diagonal C-action. For the first two maps
this follows from Proposition 6.8 and Corollary 6.9, which imply that the Thom
diagonal and T (f ) → M F (or T (f ) → M SF ) are both C-maps. For the last
map the claim follows from the fact that the multiplication H ∧ H → H is a
map of commutative symmetric ring spectra, hence in particular a C-map. 8. Symmetrization of diagram Thom spectra
In this section we first generalize the definition of the symmetric Thom spectrum functor to other types of diagram spectra. We then show how the results
in the previous sections can be used to turn such diagram Thom spectra into
symmetric spectra.
8.1. Diagram spaces and diagram Thom spectra. Given a small category
D, we define a D-space to be a functor X : D → U and we write DU for the
category of such functors. Suppose that we are given a functor φ : D → I.
Then we can generalize the notion of a symmetric spectrum by introducing the
topological category DS that has the same objects as D, but whose morphism
spaces are defined by
_
S b−α ,
DS (a, b) =
α∈D(a,b)
b−α
where S
is shorthand notation for S φ(b)−φ(α) , cf. Section 3.1. The composition is defined as for IS . We define a D-spectrum to be a continuous based
functor DS → T and we write DS T for the category of such functors. Thus, a
D-spectrum is given by a family of based spaces X(a) indexed by the objects
a in D, together with a family of based structure maps S b−α ∧ X(a) → X(b)
indexed by the morphisms α : a → b in D. It is required (i) that the structure
map associated to an identity morphism 1a : a → a is the canonical identification S 0 ∧ X(a) → X(a), and (ii) that given a pair of composable morphisms
α : a → b and β : b → c, the following diagram is commutative
S c−β ∧ S b−α ∧ X(a) −−−−→ S c−β ∧ X(b)




y
y
S c−βα ∧ X(a)
−−−−→
X(c).
In particular, if φ denotes the identity functor on I, then IS T is an alternative
notation for the category of symmetric spectra. Suppose now that D has the
structure of a strict monoidal category. As in the case of I-spaces, DU inherits
a monoidal structure from D which is symmetric monoidal if D is. If in addition
φ is strict monoidal, then the monoidal structure of D also induces a monoidal
structure on DS which is symmetric monoidal if D and φ are. This in turn
induces a monoidal structure on the category of D-spectra DS T which again is
symmetric monoidal if D and φ are. The I-space BF pulls back to a D-space
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via φ and the definition of the symmetric Thom spectrum functor immediately
generalizes to give a Thom spectrum functor
T : DU/BF → DS T .
The proof of Theorem 1.1 generalizes to show that this is a strong monoidal
functor which is symmetric monoidal if D and φ are.
8.2. Examples of diagram Thom spectra. Many examples of Thom spectra arise from compatible families of groups over the topological monoids F (n).
It often happens that such a family defines a D-diagram of groups for some
strict monoidal category D over I and if the induced maps of classifying spaces
define a D-space over BF we get an associated D-Thom spectrum. We begin
by fixing notation for some of the relevant categories. For each k ≥ 1 we have
the strict symmetric monoidal faithful functor
ψk : I → I,
n 7→ k
· · ⊔ k} .
| ⊔ ·{z
n
and we write I[k] for its image in I. Thus, I[k] is a strict symmetric monoidal
category whose objects have cardinality a multiple of k, and whose morphisms
permute blocks of k letters simultaneously. Let us write M for the subcategory of injective order preserving morphisms in I. This inherits a strict
monoidal (but not symmetric monoidal) structure from I and we similarly
define monoidal subcategories M[k] for k ≥ 1.
Example 8.1 (The classical groups). The orthogonal groups O(n) and the special orthogonal groups SO(n) define the commutative I-space monoids BO
and BSO that give rise to the commutative symmetric ring spectra M O and
M SO. The unitary groups U (n) and the special unitary groups SU (n) define
the commutative I[2]-space monoids BU and BSU that give rise to the commutative I[2]-ring spectra M U and M SU . The symplectic groups Sp(n) define
the commutative I[4]-space monoid BSp that gives rise to the commutative
I[4]-spectrum M Sp.
Example 8.2 (Discrete groups and I-spaces). The symmetric groups Σn define
the commutative I-space monoid BΣ in which the monoid structure is induced
by concatenation of permutations. This gives rise to the commutative symmetric ring spectrum M Σ whose associated bordism theory has been studied by
Bullett [9]. Other systems of discrete groups that give rise to symmetric ring
spectra include the general linear groups GLn (Z), the groups (Z/2)n of diagonal matrices with entries ±1, and the groups Σn ≀ Z/2 of permutation matrices
with entries ±1. For details and more examples, see [9] and [12].
Example 8.3 (Braid groups and M-spaces). The family of braid groups B(n)
defines an M-space monoid BB in a natural way. We refer to [5] for the
definition and basic properties of the braid groups. If we view an element of
B(n) as a system of n strings in the usual way, then the monoid structure on
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BB is induced by concatenation of such systems. Let ρ denote the sequence of
monoid homomorphisms
ρn : B(n) → Σn → F (n)
where the first map takes a system of strings to the induced permutation of the
endpoints and the second map in the canonical inclusion. This defines a map
Bρ : BB → BF of M-space monoids and we write M B for the associated MThom ring spectrum. The underlying spectrum of M B has been analyzed in [9]
and [10], where it is shown to be equivalent to the Eilenberg-Mac Lane spectrum
HZ/2. Suppose that G is an M-diagram of groups over the monoids F (n)
and that the homomorphisms ρn can be factored as B(n) → G(n) → F (n).
If the M-space BG admits a monoid structure such that the induced map
BB → BG is a map of M-space monoids over BF it then follows that M G is
an M-module spectrum over M B. For example, this applies to M Σ and M O
but not to M SO. We show how to symmetrize the constructions so as to get
symmetric Thom spectra in Section 8.3. Again we refer to [9], [10], [12] for
further examples.
Example 8.4 (Maps to BF (k) and I[k]-spaces). For our next class of examples
we need some preliminary definitions. Let X be a based space and let X • be
the I-space defined by n 7→ X n . Given a morphism α : m → n, the induced
map α∗ : X m → X n is defined by
α∗ (x1 , . . . , xm ) = (xα−1 (1) , . . . , xα−1 (n) ),
with the convention that x∅ is the base point in X. We give X • the structure
of a commutative I-space monoid using the identifications X m × X n = X m+n .
Suppose now that f : X → BF (k) is a based map. Then we view X • as an
I[k]-space via the isomorphism ψk : I → I[k], and the maps
µ
→ BF (k
· · ⊔ k})
X n → BF (k)n −
| ⊔ ·{z
n
define a map of I[k]-space monoids X • → BF , where we view BF as an
I[k]-space by restriction. We write M X ∧∞ for the associated commutative
I[k]-ring spectrum, the function f being understood. In the cases X = BO(1)
and X = BU (1), we get the Thom spectra M O(1)∧∞ and M U (1)∧∞ that
represent the bordism theories of manifolds with stable normal bundle given as
an ordered sum of real or complex line bundles. These Thom spectra have been
analyzed by Arthan and Bullett [1], [9]. Letting X = BO(k) or X = BU (k), we
similarly get the I[k]-spectrum M O(k)∧∞ and the I[2k]-spectrum M U (k)∧∞ .
8.3. Symmetrization of diagram Thom spectra. As demonstrated in the
last section, many Thom spectra naturally arise as D-Thom spectra associated
to maps of D-spaces f : X → BF for suitable monoidal categories D over I. In
the applications it is often convenient to replace such a D-Thom spectrum by
a symmetric Thom spectrum and our preferred way of doing is to first transform f to a map of I-spaces and then evaluate the symmetric Thom spectrum
functor on this transformed map. We shall discuss two ways of performing
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this “symmetrization” procedure: in this section we consider symmetrizations
using the I-space lifting functor R and in the next section we consider symmetrizations via (homotopy) Kan extension.
For simplicity we shall from now on assume that D is a monoidal subcategory
of I such that the intersection D ∩ N is a cofinal subcategory of N . Thus, any
D-spectrum has an underlying D ∩ N -spectrum and we define the spectrum
homotopy groups in the usual way by evaluating the colimit of the associated
D ∩ N -diagram of homotopy groups. Given a map f : X → BF , we write (by
abuse of notation) fhD for the composite map
f
hD
fhD : XhD −−
→ BFhD → BFhI .
Applying the I-space lifting functor R to this map we get a functor
(8.5)
DU/BF → IU/BF,
f
R(fhD )
(X −
→ BF ) 7→ (RfhD (XhD ) −−−−−→ BF ).
We say that a map of D-spaces X → Y is a D-equivalence if the induced map
XhD → YhD is a weak homotopy equivalence.
Lemma 8.6. The restriction of RfhD (XhD ) to a D-space is related to X by a
chain of D-equivalences over BF .
Proof. In analogy with the case of I-spaces considered in Section 5, there is a
diagram of D-equivalences X ← X → RfhD (XhD ) over BF .
It follows from the D-space analogue of Bökstedt’s approximation Lemma 4.4
that if X → Y is a D-equivalence of convergent D-spaces X and Y , then the
connectivity of the maps X(d) → Y (d) tends to infinity with d. The previous
lemma therefore has the following consequence.
Proposition 8.7. If X is a convergent D-space and f : X → BF is a map
of D-spaces which is level-wise T -good, then the restriction of the symmetric
spectrum T (R(fhD )) to a D-spectrum is π∗ -equivalent to T (f ).
This construction preserves multiplicative structures in the sense that if X is
a D-space monoid and f : X → BF a map of D-space monoids, then fhD is a
map of topological monoids and T (fhD ) is a symmetric ring spectra by Lemma
4.19. In the following we consider the effect of applying the construction to the
examples considered in the previous section.
Example 8.8. Let X be an I[k]-space with an action of an operad C that is
augmented over the Barratt-Eccles operad. If f : X → BF is a map of C-I[k]spaces, then the induced map
fhI[k] : XhI[k] → BFhI[k] → BFhI
is map of C-spaces and it follows from Corollary 6.9 that the symmetric spectrum T (fhI[k] ) inherits a C-action. Here we use the canonical isomorphism of
categories ψk : I → I[k] to identify XhI[k] with (ψk∗ X)hI , and we transfer the
C-action on (ψk∗ X)hI defined in Corollary 6.6 to XhI[k] via this identification.
This applies in particular to the map of commutative I[2]-spaces BU → BF
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to give a model of the Thom spectrum M U as a symmetric ring spectrum with
an action of the Barratt-Eccles operad. We shall see how to realize M U as a
strictly commutative symmetric ring spectrum in Example 8.17.
Example 8.9. Let as before BB denote the M-space monoid defined by the
braid groups. We shall identify the map BBhN → BBhM in terms of Quillen’s
plus construction. Firstly, it follows from the homological stability of the braid
groups (see [11], III, Appendix) and the homological version of Lemma 4.4,
that this map is a homology isomorphism. Secondly, the monoidal structure
of M gives BBhM the structure of a topological monoid, which in particular
implies that its fundamental group is abelian. Thus, the map in question has
the effect of abelianizing the fundamental group. The space BBhN may be
identified with the classifying space of the infinite braid group B(∞). Since
the commutator subgroup of the latter is perfect, it follows from the above that
we may identify BBhM with Quillen’s plus construction BB+
hN . It is proved
in [10] that there is a homotopy commutative diagram
θ
BBhN −−−−→ Ω2 (S 3 )


η
(Bρ)
hN
y
y
BFhN
BFhN ,
where η denotes the “Mahowald orientation”, that is, the extension of the nontrivial map S 1 → BFhN to a 2-fold loop map. It is a theorem of Mahowald
[24], that the Thom spectrum of η is stably equivalent to the Eilenberg-Mac
Lane spectrum HZ/2. By the universal property of the plus construction, we
conclude from the above that there is a homotopy commutative diagram
∼
BBhM −−−−→ Ω2 (S 3 )


η
(Bρ)
hM
y
y
BFhI
BFhI ,
where the upper map is a homotopy equivalence as indicated. Consequently,
the symmetric ring spectrum T ((Bρ)hM ) is a model of HZ/2.
Example 8.10. Let BGL(Z) be the commutative I-space monoid associated
to the general linear groups GLn (Z). As in the case of the braid groups, we
may identify BGL(Z)hN → BGL(Z)hI in terms of Quillen’s plus construction.
Indeed, by the homological stability of the groups GLn (Z), it follows that this
map is a homology equivalence. Since BGL(Z)hI is a topological monoid it has
abelian fundamental group, hence it may be identified with BGL∞ (Z)+ ; the
base point component of Quillen’s algebraic K-theory space. In similar fashion,
starting with the I-space BΣ, we may identify BΣhI with BΣ+
∞ , which by the
Barratt-Priddy-Quillen Theorem is equivalent to the base point component of
Q(S 0 ).
Example 8.11. Let f : X → BF (k) be a based map and consider the associated
map of I[k]-spaces X • → BF . It is proved in [34] that if X is well-based and
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Christian Schlichtkrull
•
connected, then XhI
is a model of the infinite loop space Q(X). Identifying
•
•
XhI with XhI[k] via the isomorphism ψk , it follows as in Example 8.8 that
•
the induced map XhI[k]
→ BFhI is a map of E-spaces which models the usual
extension of f to a map of infinite loop spaces. If we instead think of X • as an
M-space by restriction, then one can show that XhM is homotopy equivalent
•
to the colimit XM
, that is, to the free based monoid generated by X. By a
•
•
theorem of James [18] the latter is a model of ΩΣ(X), and the map XhM
→ XhI
corresponds to the inclusion of ΩΣ(X) in Q(X).
Example 8.12. Let Ek be the kth stage of the Smith filtration of the BarrattEccles operad E and write X 7→ Ek (X) for the associated monad on based
spaces, see [3], [38]. Thus, Ek is equivalent to the little k-cubes operad, and
if X is a well-based connected space, then Ek (X) is a combinatorial model of
Ωk Σk (X). Given a based map f : X → BFhI , we use that Ek is augmented
over E to extend f to a map of Ek -spaces
Ek (f ) : Ek (X) → Ek (BFhI ) → BFhI ,
which for connected X is a model of the usual extension of f to a k-fold loop
map. It follows that the associated symmetric Thom spectrum T (Ek (f )) is
equipped with an Ek -action.
8.4. Symmetrization via Kan Extension. Let again D be a monoidal subcategory of I such that D ∩ N is cofinal in N and let us write j : D → I for
the inclusion. We first consider homotopy Kan extensions along j. Recall that
given a D-space X, the homotopy Kan extension is the I-space j∗h (X) defined
by
j∗h (X)(n) = hocolim X ◦ πn ,
(j↓n)
see Appendix A.1. The functor
j∗h :
j∗h (−)
induces a functor
DU/BF → IU/BF, (f : X → BF ) 7→ (j∗h (f ) : j∗h (X) → j∗h (BF ) → BF )
which is I-equivalent to the functor (8.5) in the sense of the following lemma.
Lemma 8.13. There is a natural I-equivalence j∗h (X) → RfhD (XhD ) of Ispaces over BF .
Proof. Notice first that there is a commutative diagram
j∗h (X) −−−−→


y
BF


y
XhD −−−−→ BFhI ,
inducing a map of I-spaces j∗h (X) → RfhD (XhD ) over BF . Since this is also
a map over XhD , the result follows from the fact that j∗h (X) → XhD and
RfhD (XhD ) → XhD are I-equivalences, see Lemma A.2 and the proof of Proposition 4.9.
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We now turn to (categorical) Kan extensions. Given a D-space, the Kan extension j∗ (X) is defined as the homotopy Kan extension except that we use
the colimit instead of the homotopy colimit, that is,
j∗ (X)(n) = colim X ◦ πn .
(j↓n)
The functor j∗ is left adjoint to the functor that pulls an I-space back to a
D-space via j and it induces a functor
j∗ : DU/BF → IU/BF,
(X → BF ) 7→ (j∗ (X) → j∗ (BF ) → BF )
where the map j∗ (BF ) → BF is the counit of the adjunction. This functor is
strong monoidal and is symmetric monoidal if D and j are. Thus, in the latter
case it takes commutative D-space monoids to commutative I-space monoids.
The drawback of using the categorical Kan extension is of course that it is
homotopically well-behaved only under suitable cofibration conditions on the
D-space X and the main purpose of this section is to formulate such conditions.
More precisely, we shall consider an inclusion j : D → I of a (not necessarily
monoidal) subcategory D of I and we shall formulate conditions on D and X
which ensure that the canonical map j∗h (X) → j∗ (X) is a level-wise equivalence.
Given an object d0 of D, consider the category (D ↓ d0 ) of objects in D over
d0 and let ∂d0 be the subcategory obtained by excluding the terminal objects.
Lemma 8.14. Let j : D → I be the inclusion of a subcategory D and suppose
that X is a D-space such that the map
colim X ◦ πd0 → colim X ◦ πd0 = X(d0 )
∂d0
(D↓d0 )
is a cofibration for all objects d0 in D. Then j∗h (X) → j∗ (X) is a level-wise
equivalence.
Proof. Notice first that the category (j ↓ n) is a preorder in the sense that the
morphism sets have at most one element. Choosing a representative for each
isomorphism class we get an equivalent skeleton subcategory A(n) (in fact a
partially ordered set), and it suffices to show that the map
hocolim X ◦ πn → colim X ◦ πn
A(n)
A(n)
is a weak homotopy equivalence. The advantage of this is that the category
A(n) is very small in the sense that its nerve only has finitely many nondegenerate simplices. In this situation there is a general model categorical
criterion for comparing the homotopy colimit to the colimit, see [13], Section
10. Working in the Strøm model category on U [41], we must check that for
each object a in A(n) the map
colim X ◦ πn ◦ πa → colim X ◦ πn ◦ πa = X(πn (a))
∂a
(A(n)↓a)
is a cofibration. Here we use the notation ∂a for the subcategory of (A(n) ↓ a)
obtained by excluding the terminal object. It remains to see that if a is an
object of the form d0 → n, then this criterion is the same as that stated in the
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Christian Schlichtkrull
lemma. On the one hand we may view (A(n) ↓ a) as a skeleton subcategory of
((j ↓ n) ↓ a) and on the other hand we may identify the latter category with
(D ↓ d0 ). Taken together this gives a homeomorphism
colim X ◦ πn ◦ πa ∼
= colim X ◦ πd0
∂d0
∂a
and the conclusion follows.
The criterion in Lemma 8.14 is not very practical and in order to have a more
convenient formulation we impose conditions on the subcategory D of I. We
say that D has the intersection property if each diagram in D of the form
δ
δ
2
1
d2
d12 ←−
d1 −→
can be completed to a commutative square
(8.15)
d0 −−−−→


y
δ
d1

δ
y1
d2 −−−2−→ d12
in D such that the image of the composite morphism equals the intersection
of the images of δ1 and δ2 . For example, the monoidal subcategories I[k] and
J [k] have the intersection property for all k ≥ 1. We say that a D-space
X is intersection cofibrant if for any diagram of the form (8.15), such that
the intersection of the images of δ1 and δ2 equals the image of the composite
morphism, the induced map
X(d1 ) ∪X(d0 ) X(d2 ) → X(d12 )
is a cofibration. By Lillig’s union theorem [21] for cofibrations, this is equivalent
to the requirement that (i) any morphism d1 → d2 in D induces a cofibration
X(d1 ) → X(d2 ), and (ii) that the intersection of the images of X(d1 ) and
X(d2 ) in X(d12 ) equals the image of X(d0 ).
Proposition 8.16. Let j : D → I be the inclusion of a subcategory D which has
the intersection property and let X be a D-space which is intersection cofibrant.
Then the map j∗h (X) → j∗ (X) is a level-wise equivalence.
Proof. We show that the assumptions on D and X imply that the criterion in
Lemma 8.14 is satisfied. Given an object d0 in D, consider the range functor
r : (D ↓ d0 ) → (I ↓ d0 ) → P(d0 ),
δ
r(d −
→ d0 ) = δ(d) ⊆ d0 ,
where P(d0 ) denotes the category of subsets and inclusions in d0 . The assumption that D has the intersection property implies that the image of r is a
full subcategory of P(d0 ) that is closed under inclusions and that r defines an
equivalence of categories between (D ↓ d0 ) and its image. Thus, we might as
well view X ◦ πd0 as a diagram U 7→ X(U ) indexed on the objects U in a full
subcategory A of P(d0 ) that is closed under intersections. By assumption (i)
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above we may view X(U ) as a closed subspace of X(d0 ) for all U ∈ A and by
assumption (ii) we have the equality
X(U ) ∩ X(V ) = X(U ∩ V )
for all pairs of objects U and V in A. It therefore follows from the gluing lemma
for continuous functions on a union of closed subspaces that colim∂d0 X ◦ πd0 is
homeomorphic to the union of the subspaces X(U ) of X(d0 ) for U 6= d0 . The
conclusion then follows from Lillig’s union theorem for cofibrations [21].
Example 8.17. Let j : I[2] → I be the inclusion of the symmetric monoidal
subcategory I[2]. Since I[2] has the intersection property and the commutative
I[2]-space monoid BU is intersection cofibrant, it follows from Lemma 8.13 and
Proposition 8.16 that there is a chain of I-equivalences
∼
∼
→ R(BUhI )
j∗ (BU ) ←
− j∗h (BU ) −
over BF . Thus, it follows from Proposition 8.7 together with Theorem 1.4 and
Lemma 2.3 that applying the symmetric Thom spectrum functor to the commutative I-space monoid j∗ (BU ) gives a commutative symmetric ring spectrum
which is a model of M U .
8.5. Orthogonal Thom spectra and diagram lifting. Recall from [27]
that an orthogonal spectrum is a spectrum X such that the nth space X(n) has
an action of the orthogonal group O(n), and such that the iterated structure
maps S m ∧ X(n) → X(m + n) are O(m) × O(n)-equivariant. We write SpO
for the category of orthogonal spectra. Let V be the topological category
whose objects are the vector spaces Rn , and whose morphisms are the linear
isometries. A V-space is a continuous functor V → U, and we write VU for
the category of such functors. The symmetric monoidal structure of V induces
a symmetric monoidal structure on VU in the usual way, and the I-space
BF extends to a commutative V-space monoid. Applying the Thom space
construction level-wise as in the definition of the symmetric Thom spectrum
functor, we get the symmetric monoidal orthogonal Thom spectrum functor
T : VU/BF → SpO .
In order to construct orthogonal Thom spectra from space level data, we need
a V-space version
R : U/BFhV → VU/BF
of the I-space lifting functor. Here the homotopy colimit BFhV denotes the
realization of the simplicial space
a
[k] 7→
V(Rn1 , Rn0 ) × · · · × V(Rnk , Rnk−1 ) × BF (nk ).
n0 ,...,nk
The statement in Lemma 4.4 remains true with V instead of I, and we conclude
from this that the canonical map BFhN → BFhV is a weak homotopy equivalence. The definition of the V-space lifting functor is then completely analogous
to the definition of the I-space lifting functor in Section 4.2: Let BF be the
V-space defined by the homotopy Kan extension along the identity functor on
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Christian Schlichtkrull
V. Given a map f : X → BFhV , we define Rf (X) to be the homotopy pullback
of the diagram of V-spaces
f
π
− BF ,
X−
→ BFhV ←
and we define R(f ) to be the composition
R(f ) : Rf (X) → BF → BF.
The Barratt-Eccles operad acts on BFhV and the results on preservation of
operad actions from Section 6 carry over to this setting.
Appendix A. Homotopy colimits
We here collect the facts about homotopy colimits needed in the paper. We
shall adapt the definitions of Bousfield and Kan [7], except that we work with
topological spaces instead of simplicial sets. Thus, given a small category A
and an A-diagram X : A → U, the homotopy colimit hocolimC X is defined to
be the realization of the simplicial space
a
(A.1)
[k] 7→
X(ak ),
a0 ←···←ak
where the coproduct is over the k-simplices of the nerve N• C. It is sometimes
convenient to view this as the classifying space of the topological category A(X)
whose space of objects is the disjoint union of the spaces X(a) where a runs
through the objects of A. A morphism (a, x) → (a′ , x′ ) in A(X) is specified by
a morphism α : a → a′ in A such that α∗ x = x′ . If X is a based A-diagram,
that is, a functor X : A → T , then the inclusion of the base points gives a map
BA → BA(X) and we define the based homotopy colimit to be the quotient
space. Equivalently, this is the realization of the simplicial space obtained by
replacing the disjoint union in (A.1) by the wedge product.
A.1. Homotopy Kan extensions. Let φ : A → B be a functor between
small categories. Given an A-diagram X, the (left) homotopy Kan extension
φh∗ X is the B-diagram defined by
φh∗ X(b) = hocolim X ◦ πb .
(φ↓b)
The homotopy colimit is over the category (φ ↓ b) whose objects are pairs
(a, β) in which a is an object of A and β : φ(a) → b is a morphism in B. A
morphism (a, β) → (a′ , β ′ ) is given by a morphism α : a → a′ in A such that
β = β ′ ◦ φ(α). The functor πb : (φ ↓ b) → A is defined by (a, β) 7→ a. We
recall that the categorical Kan extension φ∗ X is defined using the categorical
colimit instead of the homotopy colimit, see [22]. If B is the terminal category
∗ and p : A → ∗ the projection, then p∗ X and ph∗ X are respectively the colimit
and the homotopy colimit of the A-diagram X. Notice, that the functors πb
define a map of B-diagrams from φh∗ X to the constant B-diagram hocolimA X.
A proof of the following well-known lemma can be found in [34].
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Lemma A.2. The induced map
∼
π : hocolim φh∗ X −
→ hocolim X.
B
A
is a weak homotopy equivalence.
This lemma may be viewed as a statement about the composition of two derived
functors. Given an additional functor ψ : B → C, one can more generally show
∼
→ (ψφ)h∗ . In the lemma
that there is a natural equivalence of functors ψ∗h φh∗ −
below, we shall consider the case where X has the form φ∗ Y for a B-diagram Y ,
and we shall relate π to the map of homotopy colimits induced by the natural
transformation of B-diagrams
t : φh∗ φ∗ Y → φ∗ φ∗ Y → Y,
where the first arrow is the canonical projection from the homotopy colimit
to the colimit and the second arrow is given by the universal property of the
categorical Kan extension.
Lemma A.3. Given a B-diagram Y , the diagram
π
/ hocolim φ∗ Y
hocolim φh∗ φ∗ Y
A
B
NNN
qq
NNN
q
q
NNN
qqq
t
NN&
xqqq φ
hocolim Y
B
is homotopy commutative by a canonical choice of a natural homotopy.
Proof. We may view the homotopy colimit of the B-diagram φh∗ φ∗ Y as the
realization of the bisimplicial space
a
φ∗ Y (aj ),
([i], [j]) 7→
n
b0 ←...←bi ←φ(a0 )
a0 ←...←aj
o
and it is well-known that this is homeomorphic to the realization of the diagonal
simplicial space. Restricting to this simplicial space, the two maps in the
diagram are induced by the simplicial maps that map a simplex
γ
α
α
i
1
ai , y)
. . . ←−
− φ(a0 ), a0 ←−
(b0 ← . . . ← bi ←
with y in φ∗ Y (ai ), to
(b0 ← . . . ← bi , γ∗ φ(α1 . . . αi )∗ y),
φ(α0 )
φ(αi )
respectively (φ(a0 ) ←−−− . . . ←−−− φ(ai ), y).
The required homotopy between the topological realizations of these maps is
then defined by
[(b0 ← . . . ← bi ← φ(a0 ) ← . . . ← φ(ai ), y); (su, (1 − s)u)],
i
for u ∈ ∆ and s ∈ I. Here I denotes the unit interval and
∆i = {(u0 , . . . , ui ) ∈ I i+1 : u0 + · · · + ui = 1}
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Christian Schlichtkrull
is the standard i-simplex.
The following lemma is needed to ensure that the functor R defined in Section
4.2 takes values in the subcategory of level-wise T -good objects in IU/BF .
Lemma A.4. Let A be a small category and let fa : Xa → BF (n) be an Adiagram in U/BF (n). If each fa classifies a well-based quasifibration, then the
induced map
f : hocolim Xa → BF (n)
A
also classifies a well-based quasifibration.
Proof. Let Wa = fa∗ V (n), and notice that f ∗ V (n) is homeomorphic to
hocolimA Wa since topological realization preserves pullback diagrams. It follows that the pullback of V (n) → BF (n) along f is homeomorphic to the
realization of the simplicial map
a
a
Wak →
Xak .
a0 ←···←ak
a0 ←···←ak
These are good simplicial spaces in the sense of [36], Appendix A, and the
map is a degree-wise quasifibration by assumption. The result then follows
from standard results on realization of simplicial quasifibrations and simplicial
cofibrations, see e.g. [36], Proposition 1.6 and [19].
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Christian Schlichtkrull
Department of Mathematics
University of Bergen
Johannes Brunsgate 12
5008 Bergen
Norway
[email protected]
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